Hypothesis Tests for Population Means Bernd Schr oder logo1 Bernd - - PowerPoint PPT Presentation

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Hypothesis Tests for Population Means

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Introduction

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Introduction

• 1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Introduction

• 1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.

• 2. The setup is always a test to reject a given null hypothesis

H0 in favor of an alternative hypothesis Ha.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Introduction

• 1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.

• 2. The setup is always a test to reject a given null hypothesis

H0 in favor of an alternative hypothesis Ha.

• 3. The probability α of a type I error

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Introduction

• 1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.

• 2. The setup is always a test to reject a given null hypothesis

H0 in favor of an alternative hypothesis Ha.

• 3. The probability α of a type I error (the significance level of

the test)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Introduction

• 1. Once the idea of hypothesis tests is understood, we want to

set up standard procedures for frequently used tests.

• 2. The setup is always a test to reject a given null hypothesis

H0 in favor of an alternative hypothesis Ha.

• 3. The probability α of a type I error (the significance level of

the test) will be given before the test.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. For H0 : µ = µ0 and Ha : µ > µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. For H0 : µ = µ0 and Ha : µ > µ0, the rejection region should be an interval (c,∞) for some c > µ0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. For H0 : µ = µ0 and Ha : µ > µ0, the rejection region should be an interval (c,∞) for some c > µ0. α = P

• X > c
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. For H0 : µ = µ0 and Ha : µ > µ0, the rejection region should be an interval (c,∞) for some c > µ0. α = P

• X > c
• = P

X − µ0 σ/√n > c− µ0 σ/√n

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. For H0 : µ = µ0 and Ha : µ > µ0, the rejection region should be an interval (c,∞) for some c > µ0. α = P

• X > c
• = P

X − µ0 σ/√n > c− µ0 σ/√n

• =

P

• Z > c− µ0

σ/√n

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. For H0 : µ = µ0 and Ha : µ > µ0, the rejection region should be an interval (c,∞) for some c > µ0. α = P

• X > c
• = P

X − µ0 σ/√n > c− µ0 σ/√n

• =

P

• Z > c− µ0

σ/√n

• zα

= c− µ0 σ/√n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. For H0 : µ = µ0 and Ha : µ > µ0, the rejection region should be an interval (c,∞) for some c > µ0. α = P

• X > c
• = P

X − µ0 σ/√n > c− µ0 σ/√n

• =

P

• Z > c− µ0

σ/√n

• zα

= c− µ0 σ/√n is the cutoff point.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. For H0 : µ = µ0 and Ha : µ > µ0, the rejection region should be an interval (c,∞) for some c > µ0. α = P

• X > c
• = P

X − µ0 σ/√n > c− µ0 σ/√n

• =

P

• Z > c− µ0

σ/√n

• zα

= c− µ0 σ/√n is the cutoff point. For Ha : µ < µ0 or Ha : µ = µ0 computations are similar.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)

✲ µ0 α

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)

✲ µ0 α Upper tail test for µ≤µ0:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)

✲ µ0 α Upper tail test for µ≤µ0: Tail probability is ≤α (small) if µ≤µ0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)

✲ µ0 α

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)

✲ µ0 α Lower tail test for µ≥µ0:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)

✲ µ0 α Lower tail test for µ≥µ0: Tail probability is ≤α (small) if µ≥µ0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

✲ µ0

α 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

✲ µ0

α 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

✲ µ0

α 2 α 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

✲ µ0

α 2 α 2

Two tailed test for µ = µ0:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Consider a random sample from a normal population with known standard deviation σ. To test the null hypothesis µ = µ0 against various alternative hypotheses, we use the test statistic z = x− µ0 σ/√n and define the following rejection regions.

• 1. For Ha : µ > µ0 use z ≥ zα (upper tailed test)
• 2. For Ha : µ < µ0 use z ≤ −zα (lower tailed test)
• 3. For Ha : µ = µ0 use |z| ≥ z α

2 (two-tailed test)

✲ µ0

α 2 α 2

Two tailed test for µ = µ0: Tail probability is α (small) if µ=µ0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests

• 1. Determine the parameter of interest.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests

• 1. Determine the parameter of interest.
• 2. Determine the null hypothesis H0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests

• 1. Determine the parameter of interest.
• 2. Determine the null hypothesis H0.
• 3. Determine the alternative hypothesis Ha.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests

• 1. Determine the parameter of interest.
• 2. Determine the null hypothesis H0.
• 3. Determine the alternative hypothesis Ha.
• 4. Choose the appropriate test statistic.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests

• 1. Determine the parameter of interest.
• 2. Determine the null hypothesis H0.
• 3. Determine the alternative hypothesis Ha.
• 4. Choose the appropriate test statistic.
• 5. Determine the rejection region using the significance level.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests

• 1. Determine the parameter of interest.
• 2. Determine the null hypothesis H0.
• 3. Determine the alternative hypothesis Ha.
• 4. Choose the appropriate test statistic.
• 5. Determine the rejection region using the significance level.
• 6. Determine if the test statistic falls into the rejection region
• r not.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Performing Hypothesis Tests

• 1. Determine the parameter of interest.
• 2. Determine the null hypothesis H0.
• 3. Determine the alternative hypothesis Ha.
• 4. Choose the appropriate test statistic.
• 5. Determine the rejection region using the significance level.
• 6. Determine if the test statistic falls into the rejection region
• r not. Reject or don’t reject accordingly.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest: µ, the true mean displacement.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest: µ, the true mean displacement. Null hypothesis:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. A new type of body armor is tested if it satisfies the

specification of at most µ0 = 1.9in of displacement when hit with a certain type of bullet. The manufacturer tests by firing

• ne round each at 36 samples of the new armor and measuring

the displacement upon impact. The result is a sample mean displacement of 1.91 in. Assume the displacements are normally distributed with mean µ and a standard deviation of 0.06 in. Test if the armor is up to specifications at the 10% significance level. Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region: We use an upper tailed test and the rejection region is z > z0.1 ≈ 1.28.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region: We use an upper tailed test and the rejection region is z > z0.1 ≈ 1.28. Substitute values into test statistic:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region: We use an upper tailed test and the rejection region is z > z0.1 ≈ 1.28. Substitute values into test statistic: z

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region: We use an upper tailed test and the rejection region is z > z0.1 ≈ 1.28. Substitute values into test statistic: z = x− µ0 σ/√n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region: We use an upper tailed test and the rejection region is z > z0.1 ≈ 1.28. Substitute values into test statistic: z = x− µ0 σ/√n = 1.91−1.9 0.06/ √ 36

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region: We use an upper tailed test and the rejection region is z > z0.1 ≈ 1.28. Substitute values into test statistic: z = x− µ0 σ/√n = 1.91−1.9 0.06/ √ 36 = 0.01 0.01

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region: We use an upper tailed test and the rejection region is z > z0.1 ≈ 1.28. Substitute values into test statistic: z = x− µ0 σ/√n = 1.91−1.9 0.06/ √ 36 = 0.01 0.01 = 1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region: We use an upper tailed test and the rejection region is z > z0.1 ≈ 1.28. Substitute values into test statistic: z = x− µ0 σ/√n = 1.91−1.9 0.06/ √ 36 = 0.01 0.01 = 1 Decide if to reject or not.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Parameter of interest: µ, the true mean displacement. Null hypothesis: H0 : µ = 1.9in (null value of µ0) Alternative hypothesis: Ha : µ > 1.9in (the main concern is a displacement that is larger than what is claimed). Test statistic: z = x− µ0 σ/√n = x−1.9 0.06/√n Rejection region: We use an upper tailed test and the rejection region is z > z0.1 ≈ 1.28. Substitute values into test statistic: z = x− µ0 σ/√n = 1.91−1.9 0.06/ √ 36 = 0.01 0.01 = 1 Decide if to reject or not. Do not reject the null hypothesis, because the test statistic is not in the rejection region.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Discussion

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Discussion

A type II error is more serious than a type I error here.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Discussion

A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Discussion

A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works. Otherwise false negatives may keep you from ever producing.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Discussion

A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works. Otherwise false negatives may keep you from ever producing. But when a test says “reject”, you really reject.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Discussion

A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works. Otherwise false negatives may keep you from ever producing. But when a test says “reject”, you really reject. That’s it.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Discussion

A type II error is more serious than a type I error here. At the same time, it is sensible to use a null hypothesis that says your product works. Otherwise false negatives may keep you from ever producing. But when a test says “reject”, you really reject. That’s it. Even if it’s expensive.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• =

Φ

• zα + µ0 − µ′

σ/√n

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• =

Φ

• zα + µ0 − µ′

σ/√n

• ✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• =

Φ

• zα + µ0 − µ′

σ/√n

• ✲

µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• =

Φ

• zα + µ0 − µ′

σ/√n

• ✲

µ0 α

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• =

Φ

• zα + µ0 − µ′

σ/√n

• ✲

µ0 α

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• =

Φ

• zα + µ0 − µ′

σ/√n

• ✲

µ0 α µ’

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• =

Φ

• zα + µ0 − µ′

σ/√n

• ✲

µ0 α µ’ β

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• =

Φ

• zα + µ0 − µ′

σ/√n

• ✲

µ0 α µ’ β Upper tail test for µ≤µ0:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Probability of a type II error for Ha : µ > µ0.

β(µ′) = P

• X < µ0 +zασ/√n|µ = µ′

= P X − µ′ σ/√n < zα + µ0 − µ′ σ/√n

• =

Φ

• zα + µ0 − µ′

σ/√n

• ✲

µ0 α µ’ β Upper tail test for µ≤µ0: If µ=µ’>µ0, non-rejection probability is β.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size that will yield these probabilities,

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size that will yield these probabilities, provided that a value µ′ for which β(µ′) = β is given.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size that will yield these probabilities, provided that a value µ′ for which β(µ′) = β is given. For an upper tailed test, we should choose n to satisfy the following.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size that will yield these probabilities, provided that a value µ′ for which β(µ′) = β is given. For an upper tailed test, we should choose n to satisfy the following. Φ

• zα + µ0 − µ′

σ/√n

• =

β

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size that will yield these probabilities, provided that a value µ′ for which β(µ′) = β is given. For an upper tailed test, we should choose n to satisfy the following. Φ

• zα + µ0 − µ′

σ/√n

• =

β zα + µ0 − µ′ σ/√n = −zβ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size that will yield these probabilities, provided that a value µ′ for which β(µ′) = β is given. For an upper tailed test, we should choose n to satisfy the following. Φ

• zα + µ0 − µ′

σ/√n

• =

β zα + µ0 − µ′ σ/√n = −zβ zα +zβ = −√nµ0 − µ′ σ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size that will yield these probabilities, provided that a value µ′ for which β(µ′) = β is given. For an upper tailed test, we should choose n to satisfy the following. Φ

• zα + µ0 − µ′

σ/√n

• =

β zα + µ0 − µ′ σ/√n = −zβ zα +zβ = −√nµ0 − µ′ σ √n = σ(zα +zβ) µ′ − µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

With α and β specified, we can determine a sample size that will yield these probabilities, provided that a value µ′ for which β(µ′) = β is given. For an upper tailed test, we should choose n to satisfy the following. Φ

• zα + µ0 − µ′

σ/√n

• =

β zα + µ0 − µ′ σ/√n = −zβ zα +zβ = −√nµ0 − µ′ σ √n = σ(zα +zβ) µ′ − µ0 n = σ(zα +zβ) µ′ − µ0 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of bullet.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in), normal distribution and σ = 0.06 in

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in), normal distribution and σ = 0.06 in a test at the 10% significance level has at most a chance of 0.0005 = 0.05% for a type II error.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in), normal distribution and σ = 0.06 in a test at the 10% significance level has at most a chance of 0.0005 = 0.05% for a type II error. z0.1 = 1.28

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in), normal distribution and σ = 0.06 in a test at the 10% significance level has at most a chance of 0.0005 = 0.05% for a type II error. z0.1 = 1.28 z0.0005 = 3.29

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in), normal distribution and σ = 0.06 in a test at the 10% significance level has at most a chance of 0.0005 = 0.05% for a type II error. z0.1 = 1.28 z0.0005 = 3.29 n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in), normal distribution and σ = 0.06 in a test at the 10% significance level has at most a chance of 0.0005 = 0.05% for a type II error. z0.1 = 1.28 z0.0005 = 3.29 n = 0.06(1.28+3.29) 2−1.9 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in), normal distribution and σ = 0.06 in a test at the 10% significance level has at most a chance of 0.0005 = 0.05% for a type II error. z0.1 = 1.28 z0.0005 = 3.29 n = 0.06(1.28+3.29) 2−1.9 2 =

• (0.6·4.57)2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in), normal distribution and σ = 0.06 in a test at the 10% significance level has at most a chance of 0.0005 = 0.05% for a type II error. z0.1 = 1.28 z0.0005 = 3.29 n = 0.06(1.28+3.29) 2−1.9 2 =

• (0.6·4.57)2

= ⌈7.52⌉

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. Suppose a production run of a new type of body

armor is tested if it satisfies the specification of at most µ0 = 1.9 in of displacement when hit with a certain type of

• bullet. Compute n so that with µ′ = 2 in (military spec:

displacement no more than 2 in), normal distribution and σ = 0.06 in a test at the 10% significance level has at most a chance of 0.0005 = 0.05% for a type II error. z0.1 = 1.28 z0.0005 = 3.29 n = 0.06(1.28+3.29) 2−1.9 2 =

• (0.6·4.57)2

= ⌈7.52⌉ = 8

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 1. Usually, σ is not known.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 1. Usually, σ is not known.
• 2. For large sample size, use the same tests, but use the

sample statistic z = x− µ0 s/√n .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 1. Usually, σ is not known.
• 2. For large sample size, use the same tests, but use the

sample statistic z = x− µ0 s/√n . This can be done because, by the Central Limit Theorem, for large enough n, the random variable Z = X − µ S/√n is approximately normally distributed.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 1. Usually, σ is not known.
• 2. For large sample size, use the same tests, but use the

sample statistic z = x− µ0 s/√n . This can be done because, by the Central Limit Theorem, for large enough n, the random variable Z = X − µ S/√n is approximately normally distributed. Rule of thumb: n > 40.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 1. Usually, σ is not known.
• 2. For large sample size, use the same tests, but use the

sample statistic z = x− µ0 s/√n . This can be done because, by the Central Limit Theorem, for large enough n, the random variable Z = X − µ S/√n is approximately normally distributed. Rule of thumb: n > 40. Computations and formulas are the same.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 1. Usually, σ is not known.
• 2. For large sample size, use the same tests, but use the

sample statistic z = x− µ0 s/√n . This can be done because, by the Central Limit Theorem, for large enough n, the random variable Z = X − µ S/√n is approximately normally distributed. Rule of thumb: n > 40. Computations and formulas are the same. For β and sample size, we would need to assume we know σ.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 3. For small samples from a normal distribution, use the
• ne-sample t-test.

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 3. For small samples from a normal distribution, use the
• ne-sample t-test.

Null hypothesis: H0 : µ = µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 3. For small samples from a normal distribution, use the
• ne-sample t-test.

Null hypothesis: H0 : µ = µ0 Test statistic: t = x− µ0 s/√n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 3. For small samples from a normal distribution, use the
• ne-sample t-test.

Null hypothesis: H0 : µ = µ0 Test statistic: t = x− µ0 s/√n Alternative hypotheses and rejection regions:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 3. For small samples from a normal distribution, use the
• ne-sample t-test.

Null hypothesis: H0 : µ = µ0 Test statistic: t = x− µ0 s/√n Alternative hypotheses and rejection regions:

◮ Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 3. For small samples from a normal distribution, use the
• ne-sample t-test.

Null hypothesis: H0 : µ = µ0 Test statistic: t = x− µ0 s/√n Alternative hypotheses and rejection regions:

◮ Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test) ◮ Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 3. For small samples from a normal distribution, use the
• ne-sample t-test.

Null hypothesis: H0 : µ = µ0 Test statistic: t = x− µ0 s/√n Alternative hypotheses and rejection regions:

◮ Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test) ◮ Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test) ◮ Ha : µ = µ0, t ≤ −t α 2 ,n−1 or t ≥ t α 2 ,n−1 (two tailed test) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Adjusting Test Statistics for Sample Size

• 3. For small samples from a normal distribution, use the
• ne-sample t-test.

Null hypothesis: H0 : µ = µ0 Test statistic: t = x− µ0 s/√n Alternative hypotheses and rejection regions:

◮ Ha : µ > µ0, t ≥ tα,n−1 (upper tailed test) ◮ Ha : µ < µ0, t ≤ tα,n−1 (lower tailed test) ◮ Ha : µ = µ0, t ≤ −t α 2 ,n−1 or t ≥ t α 2 ,n−1 (two tailed test)

Alternative hypotheses and rejection regions are often tabulated in statistics texts for more convenient reference.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests

• 1. Computing β(µ′) for t-tests is complicated

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests

• 1. Computing β(µ′) for t-tests is complicated, because µ and

σ are both unknown.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests

• 1. Computing β(µ′) for t-tests is complicated, because µ and

σ are both unknown. To get β(µ′), compute d = |µ0 − µ′| σ with some estimate for σ.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests

• 1. Computing β(µ′) for t-tests is complicated, because µ and

σ are both unknown. To get β(µ′), compute d = |µ0 − µ′| σ with some estimate for σ. Then use the β curves (in your statistics book) with the right number of degrees of freedom to get β.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests

• 1. Computing β(µ′) for t-tests is complicated, because µ and

σ are both unknown. To get β(µ′), compute d = |µ0 − µ′| σ with some estimate for σ. Then use the β curves (in your statistics book) with the right number of degrees of freedom to get β.

• 2. To choose the sample size, find d and your desired β.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Power and Sample Size for t-tests

• 1. Computing β(µ′) for t-tests is complicated, because µ and

σ are both unknown. To get β(µ′), compute d = |µ0 − µ′| σ with some estimate for σ. Then use the β curves (in your statistics book) with the right number of degrees of freedom to get β.

• 2. To choose the sample size, find d and your desired β.

Estimate which curve (determined by the degrees of freedom) should be at (d,β).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

Example.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a)

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon.

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100 Ha : µ = 100 (Could use µ < 100. Definitely not µ > 100.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100 Ha : µ = 100 (Could use µ < 100. Definitely not µ > 100.) Test statistic: T = x− µ0 s/√n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100 Ha : µ = 100 (Could use µ < 100. Definitely not µ > 100.) Test statistic: T = x− µ0 s/√n Rejection region: For α = 0.05, |T| > t0.025,11 ≈ 2.201

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100 Ha : µ = 100 (Could use µ < 100. Definitely not µ > 100.) Test statistic: T = x− µ0 s/√n Rejection region: For α = 0.05, |T| > t0.025,11 ≈ 2.201 Values

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100 Ha : µ = 100 (Could use µ < 100. Definitely not µ > 100.) Test statistic: T = x− µ0 s/√n Rejection region: For α = 0.05, |T| > t0.025,11 ≈ 2.201 Values: x = 98.375

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100 Ha : µ = 100 (Could use µ < 100. Definitely not µ > 100.) Test statistic: T = x− µ0 s/√n Rejection region: For α = 0.05, |T| > t0.025,11 ≈ 2.201 Values: x = 98.375, s ≈ 6.109

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100 Ha : µ = 100 (Could use µ < 100. Definitely not µ > 100.) Test statistic: T = x− µ0 s/√n Rejection region: For α = 0.05, |T| > t0.025,11 ≈ 2.201 Values: x = 98.375, s ≈ 6.109, s/√n ≈ 1.764

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100 Ha : µ = 100 (Could use µ < 100. Definitely not µ > 100.) Test statistic: T = x− µ0 s/√n Rejection region: For α = 0.05, |T| > t0.025,11 ≈ 2.201 Values: x = 98.375, s ≈ 6.109, s/√n ≈ 1.764, T = −0.921

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 105.6, 90.9, 91.2, 96.9, 96.5, 91.3, 100.1, 105.0, 99.6, 107.7, 103.3, 92.4. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypothesis at the 0.05 significance level. Parameter: mean reading µ H0 : µ = 100 Ha : µ = 100 (Could use µ < 100. Definitely not µ > 100.) Test statistic: T = x− µ0 s/√n Rejection region: For α = 0.05, |T| > t0.025,11 ≈ 2.201 Values: x = 98.375, s ≈ 6.109, s/√n ≈ 1.764, T = −0.921 Decision: Don’t reject.

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon.

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. Suppose that prior to the experiment, a value of σ = 7.5 pCi/L had been assumed.

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. Suppose that prior to the experiment, a value of σ = 7.5 pCi/L had been assumed. How many determinations would then have been appropriate to obtain β = 0.1 for the alternative µ = 95?

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. Suppose that prior to the experiment, a value of σ = 7.5 pCi/L had been assumed. How many determinations would then have been appropriate to obtain β = 0.1 for the alternative µ = 95? d = |µ′ − µ0| σ

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. Suppose that prior to the experiment, a value of σ = 7.5 pCi/L had been assumed. How many determinations would then have been appropriate to obtain β = 0.1 for the alternative µ = 95? d = |µ′ − µ0| σ = 2 3

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. Suppose that prior to the experiment, a value of σ = 7.5 pCi/L had been assumed. How many determinations would then have been appropriate to obtain β = 0.1 for the alternative µ = 95? d = |µ′ − µ0| σ = 2 3, α = 0.05

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• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. Suppose that prior to the experiment, a value of σ = 7.5 pCi/L had been assumed. How many determinations would then have been appropriate to obtain β = 0.1 for the alternative µ = 95? d = |µ′ − µ0| σ = 2 3, α = 0.05 n−1 ≈ 39

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means

logo1 Testing Normal Distributions Example Sample Size Determination t-tests

• Example. (DeVore, Section 8.2, nr. 32a) A sample of 12 Radon

detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. Suppose that prior to the experiment, a value of σ = 7.5 pCi/L had been assumed. How many determinations would then have been appropriate to obtain β = 0.1 for the alternative µ = 95? d = |µ′ − µ0| σ = 2 3, α = 0.05 n−1 ≈ 39 (tables are wonderful)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Means