Outline Significance Levels (8.2.4) z -Tests (8.2.5) ests (8 5) - - PDF document

1/18/2007 1

219323 Probability y and Statistics for Software and Knowledge Engineers

Lecture 9: Hypothesis Testing (cont.)

Monchai Sopitkamon, Ph.D.

Outline

Significance Levels (8.2.4) z-Tests (8.2.5)

ests (8 5)

Summary (8.3)

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Significance Levels I (8.2.4)

The null hypothesis H0 is rejected if

The null hypothesis H0 is rejected if the p-value is smaller than size α (significance level), and H0 is accepted if the p-value is larger than α.

Significance Levels II (8.2.4)

Error classification for hypothesis tests

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Two-Sided Hypothesis Test for a Population Mean

( )

s x n t μ − =

Size α tw o- sided t-test

Two-Sided Hypothesis Test for a Population Mean: Example

Sample size n = 18 observations

Table III provides critical points as in the left Table. Consider the test statistics |t| = 3.24, 1.625, 1.74≤|t|≤2.898

α tα/2,17 0.10 1.740 0.05 2.110 0.01 2.898

Excel sheet

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Relationship Between Confidence Intervals & Hypothesis Tests I

The value µ0 is contained within a 1 - α

µ0 level two-sided CI

if the p-value for the two-sided hypothesis test H0 : µ = µ0 versus HA : µ ≠ µ0

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + −

− −

n s t x n s t x

n n 1 , 2 / 1 , 2 /

,

α α

A

is larger than α. So, if µ0 is contained within the 1 - α level CI, the hypothesis test with size α accepts the null hypothesis, and if µ0 is contained outside the 1 - α level CI, the hypothesis test rejects H0

Relationship Between Confidence Intervals & Hypothesis Tests II

Relationship betw een hypothesis testing and confidence intervals for tw o- sided problem s

1/18/2007 5 Relationship Between Confidence Intervals & Hypothesis Tests III: Ex.47 pg.363

With t0.005, 29 = 2.756 (from Table III), a

99% two-sided t-interval for the mean tensile strength is Since µ0 = 40.0 is not contained within this CI which is consistent with the

( )

67 . 39 , 36 . 37 30 299 . 2 756 . 2 518 . 38 , 30 299 . 2 756 . 2 518 . 38 ,

1 , 2 / 1 , 2 /

= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × + × − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + −

− −

n s t x n s t x

n n α α

this CI, which is consistent with the hypothesis test H0 : µ = 40.0 versus HA : µ ≠ 40.0 having a p-value of 0.0014, so that the null hypothesis is rejected at size α = 0.01.

Excel sheet

Relationship Between Confidence Intervals & Hypothesis Tests III: Ex.14 pg.363

With t0.05, 59 = 1.671, a 90% two-sided t-

interval for the mean cylinder diameter is Since µ0 = 50.0 is contained within this CI, which is consistent with the hypothesis test H : µ = 50 0 versus

( )

028 . 50 , 970 . 49 60 134 . 671 . 1 999 . 49 , 60 134 . 671 . 1 999 . 49 ,

1 , 2 / 1 , 2 /

= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × + × − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + −

− −

n s t x n s t x

n n α α

hypothesis test H0 : µ = 50.0 versus HA : µ ≠ 50.0 having a p-value of 0.954, so that the null hypothesis is accepted at size α = 0.1.

Excel sheet

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One-Sided Hypothesis Test for a Population Mean I

( )

s x n t μ − =

Size α one- sided t-test

One-Sided Hypothesis Test for a Population Mean II

( )

s x n t μ − =

Size α one- sided t-test

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Relationship Between Confidence Intervals & Hypothesis Tests I

Relationship betw een hypothesis testing and confidence intervals for one- sided problem s

Relationship Between Confidence Intervals & Hypothesis Tests II

Relationship betw een hypothesis testing and confidence intervals for one- sided problem s

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Summarization of relationships between CIs, p-values, and significance levels (α) for two- sided and one-sided problems

One-Sided Problem Example: Ex.48 pg.366: Car Fuel Efficiency

The one-sided hypothesis are

H0 : µ ≥ 35.0 versus HA : µ < 35.0 Since the t-statistic = -1.119 > t-critical -t0.01, 19 =

• 1.328, a size α = 0.10 hypothesis test accepts

the null hypothesis. This is also consistent w/ the previous analysis where the p-value = 0.1386 > α = 0.10. Besides, the one-sided 90% t-interval

⎞ ⎛ ⎞ ⎛

contains the value µ0 = 35.0, as expected.

( )

14 . 35 , 20 915 . 2 328 . 1 271 . 34 , ,

1 ,

∞ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × + ∞ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∞ − ∈

−

n s t x

n α

μ

Excel sheet

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Power Levels

Significance level α prob that null

g p hypothesis is rejected when it is true (Type I error)

Small significance levels are

employed in hypothesis tests so that the prob of Type I error is small. T pe II error (prob that n ll

Type II error (prob that null

hypothesis is accepted when it is false) should also be minimized, thus the introduction of the “Power of a Hypothesis Test” concept.

Power of a Hypothesis Test

Power = 1 – (prob of Type II error)

Power 1 (prob of Type II error) = prob that the null hypothesis is rejected when it is false.

The larger the power value, the

better the experiment For a fi ed significance le el the

For a fixed significance level α, the

power of a hypothesis test increases as the sample size n increases.

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Outline

Significance Levels (8.2.4) z-Tests (8.2.5)

z Tests (8.2.5)

Summary (8.3)

z-Tests (8.2.5)

Used when the population SD σ is

Used when the population SD σ is known, rather than the sample SD s.

Uses z-statistic:

( )

σ μ0 − = x n z

which has standard normal dist when µ = µ0

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Two-Sided z-Test I

The p-value for the two-sided hypothesis testing problem testing problem H0 : µ = µ0 versus HA : µ ≠ µ0 based on a data set of n observations w/ a sample mean and a population SD σ is p-value = 2 x Φ(-|z|) where the Φ(x) is standard normal CDF

x

and which is known as the z-statistic.

( )

σ μ0 − = x n z

Two-Sided z-Test II

A size α test rejects the null hypothesis H0 if

the test statistic |z| falls in the rejection region |z| > zα/2 and accepts the null hypothesis H0 if the test statistic |z| falls in the acceptance region |z| ≤ zα/2 the 1 - α level two-sided CI

⎞ ⎛

consists of the values µ0 for which this hypothesis testing problem has a p-value > α,

• r the values µ0 for which the size α

hypothesis test accepts the null hypothesis.

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − ∈ n z x n z x σ σ μ

α α 2 / 2 /

,

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One-Sided z-Test I (H0: µ≤µ0)

The p-value for the two-sided hypothesis testing problem H0 : µ ≤ µ0 versus HA : µ > µ0 H0 : µ ≤ µ0 versus HA : µ µ0 based on a data set of n observations w/ a sample mean and a population SD σ is p-value = 1 – Φ(z) A size α test rejects the null hypothesis when z > zα and accepts the null hypothesis when z ≤ zα

x

⎞ ⎛

z zα the 1 - α level one-sided CI consists of the values µ0 for which this hypothesis testing problem has a p-value > α,

• r the values µ0 for which the size α

hypothesis test accepts the null hypothesis

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∞ − ∈ , n z x σ μ

α

One-Sided z-Test II (H0: µ≥µ0)

The p-value for the two-sided hypothesis testing problem H0 : µ ≥ µ0 versus HA : µ < µ0 H0 : µ ≥ µ0 versus HA : µ µ0 based on a data set of n observations w/ a sample mean and a population SD σ is p-value = Φ(z) A size α test rejects the null hypothesis when z < -zα and accepts the null hypothesis when z ≥ -zα

x

⎞ ⎛

z zα the 1 - α level one-sided CI consists of the values µ0 for which this hypothesis testing problem has a p-value > α,

• r the values µ0 for which the size α

hypothesis test accepts the null hypothesis

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∞ − ∈ n z x σ μ

α

,

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Outline

Significance Levels (8.2.4) z-Tests (8.2.5)

z Tests (8.2.5)

Summary (8.3)

Summary I (8.3)

Decision process for inferences on a population m ean

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Summary II (8.3)

Sum m ary of the t-procedure

Summary III (8.3)

Sum m ary of the z-procedure