SLIDE 1
Chapter 8
Inferences Based on a Single Sample: Tests of Hypothesis
SLIDE 2 The Elements of a Test of Hypothesis
7 elements
- 1. The Null hypothesis
- 2. The alternate, or research hypothesis
- 3. The test statistic
- 4. The rejection region
- 5. The assumptions
- 6. The Experiment and test statistic calculation
- 7. The Conclusion
SLIDE 3
The Elements of a Test of Hypothesis
Does a manufacturer’s pipe meet building code? Null hypothesis – Pipe does not meet code (H0): < 2400 Alternate hypothesis – Pipe meets specifications (Ha): > 2400
SLIDE 4 The Elements of a Test of Hypothesis
Test statistic to be used Rejection region
Determined by Type I error, which is the probability of rejecting the null hypothesis when it is true, which is . Here, we set =.05
Region is z>1.645, from z value table
n x x z
x
2400 2400
SLIDE 5 The Elements of a Test of Hypothesis
Assume that s is a good approximation of Sample of 60 taken, , s=200 Test statistic is Test statistic lies in rejection region, therefore we reject H0 and accept Ha that the pipe meets building code
12 . 2 28 . 28 60 50 200 2400 2460 2400 n s x z
2460 x
SLIDE 6 The Elements of a Test of Hypothesis
Type I vs Type II Error
Conclusions and Consequences for a Test of Hypothesis True State of Nature Conclusion H0 True Ha True Accept H0 (Assume H0 True) Correct decision Type II error (probability ) Reject H0 (Assume Ha True) Type I error (probability ) Correct decision
SLIDE 7 The Elements of a Test of Hypothesis
- 1. The Null hypothesis – the status quo. What
we will accept unless proven otherwise. Stated as H0: parameter = value
- 2. The Alternative (research) hypothesis (Ha) –
theory that contradicts H0. Will be accepted if there is evidence to establish its truth
- 3. Test Statistic – sample statistic used to
determine whether or not to reject Ho and accept Ha
SLIDE 8
The Elements of a Test of Hypothesis
4. The rejection region – the region that will lead to H0 being rejected and Ha accepted. Set to minimize the likelihood of a Type I error 5. The assumptions – clear statements about the population being sampled 6. The Experiment and test statistic calculation – performance of sampling and calculation of value of test statistic 7. The Conclusion – decision to (not) reject H0, based on a comparison of test statistic to rejection region
SLIDE 9 Large-Sample Test of Hypothesis about a Population Mean
Null hypothesis is the status quo, expressed in one
H0: = 2400 H0: ≤ 2400 H0: ≥ 2400 It represents what must be accepted if the alternative hypothesis is not accepted as a result
SLIDE 10
Large-Sample Test of Hypothesis about a Population Mean
Alternative hypothesis can take one of 3 forms:
One-tailed, upper tail Ha: <2400 One-tailed, upper tail Ha: >2400 Two-tailed Ha: 2400
SLIDE 11 Large-Sample Test of Hypothesis about a Population Mean
Rejection Regions for Common Values of Alternative Hypotheses Lower-Tailed Upper-Tailed Two-Tailed = .10 z < -1.28 z > 1.28 z < -1.645 or z > 1.645 = .05 z < -1.645 z > 1.645 Z < -1.96 or z > 1.96 = .01 z < -2.33 z > 2.33 Z < -2.575 or z > 2.575
SLIDE 12 Large-Sample Test of Hypothesis about a Population Mean
If we have: n=100, = 11.85, s = .5, and we want to test if 12 with a 99% confidence level, our setup would be as follows: H0: = 12 Ha: 12 Test statistic Rejection region z < -2.575 or z > 2.575 (two-tailed)
x
x
x z 12
SLIDE 13 Large-Sample Test of Hypothesis about a Population Mean
CLT applies, therefore no assumptions about population are needed Solve Since z falls in the rejection region, we conclude that at .01 level of significance the
- bserved mean differs significantly from 12
3 . 10 5 . 15 . 10 12 85 . 11 100 12 85 . 11 12 12 s n x x z
x
SLIDE 14
Observed Significance Levels: p- Values
The p-value, or observed significance level, is the smallest that can be set that will result in the research hypothesis being accepted.
SLIDE 15 Observed Significance Levels: p- Values
Steps: Determine value of test statistic z The p-value is the area to the right of z if Ha is one-tailed, upper tailed The p-value is the area to the left of z if Ha is
The p-valued is twice the tail area beyond z if Ha is two-tailed.
SLIDE 16
Observed Significance Levels: p- Values
When p-values are used, results are reported by setting the maximum you are willing to tolerate, and comparing p-value to that to reject or not reject H0
SLIDE 17 Small-Sample Test of Hypothesis about a Population Mean
When sample size is small (<30) we use a different sampling distribution for determining the rejection region and we calculate a different test statistic The t-statistic and t distribution are used in cases
- f a small sample test of hypothesis about
All steps of the test are the same, and an assumption about the population distribution is now necessary, since CLT does not apply
SLIDE 18 Small-Sample Test of Hypothesis about a Population Mean
where t and t/2 are based on (n-1) degrees of freedom Rejection region: Rejection region: (or when Ha: Test Statistic: Test Statistic: Ha: Ha: (or Ha: ) H0: H0: Two-Tailed Test One-Tailed Test
Small-Sample Test of Hypothesis about
n s x t
n s x t
t t
t t
2
t t
SLIDE 19 Large-Sample Test of Hypothesis about a Population Proportion
Rejection region: Rejection region: (or when where, according to H0, and Test Statistic: Test Statistic: Ha: Ha: (or Ha: ) H0: H0: Two-Tailed Test One-Tailed Test
Large-Sample Test of Hypothesis about
p p
p
p p
p
p p z
ˆ
ˆ
p p p p
p
p p z ˆ
z z
z z
2
z z
p p p p
n q p
p ˆ
1 p q
SLIDE 20 Large-Sample Test of Hypothesis about a Population Proportion
Assumptions needed for a Valid Large-Sample Test of Hypothesis for p
- A random sample is selected from a binomial
population
- The sample size n is large (condition satisfied if
falls between 0 and 1
p
p
ˆ
3
SLIDE 21 Calculating Type II Error Probabilities: More about
Type II error is associated with , which is the probability that we will accept H0 when Ha is true Calculating a value for can only be done if we assume a true value for There is a different value of for every value
SLIDE 22 Calculating Type II Error Probabilities: More about
Steps for calculating for a Large-Sample Test about 1. Calculate the value(s) of corresponding to the borders of the rejection region using one of the following:
Upper-tailed test: Lower-tailed test: Two-tailed test: x
n s z z x
x
n s z z x
x
n s z z x
x L
n s z z x
x U
SLIDE 23 Calculating Type II Error Probabilities: More about
2. Specify the value of in Ha for which is to be calculated. 3. Convert border values of to z values using the mean , and the formula 4. Sketch the alternate distribution, shade the area in the acceptance region and use the z statistics and table to find the shaded area,
a
x a
x z
x
a
SLIDE 24
Calculating Type II Error Probabilities: More about
The Power of a test – the probability that the test will correctly lead to the rejection of H0 for a particular value of in Ha. Power is calculated as 1- .
SLIDE 25 Tests of Hypothesis about a Population Variance
Hypotheses about the variance use the Chi- Square distribution and statistic The quantity has a sampling distribution that follows the chi-square distribution assuming the population the sample is drawn from is normally distributed.
2 2
1 s n
SLIDE 26 Tests of Hypothesis about a Population Variance
where is the hypothesized variance and the distribution of is based
- n (n-1) degrees of freedom
Rejection region: Or Rejection region: (or when Ha: Test Statistic: Test Statistic: Ha: Ha: (or Ha: ) H0: H0: Two-Tailed Test One-Tailed Test
Small-Sample Test of Hypothesis about
2 2
2
2 2
2 2 2
1 s n
1
2 2 2 2
2 2
2 2
2 2 2
1 s n
2 2
2 1
2 2
2
2 2
2 2
2
2
