Chapter 7: Hypothesis Testing
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Hypothesis Testing

A statistical hypothesis is a statement about the value of a population parameter.

The Alternative Hypothesis, Ha, is usually the hypothesis for which the researchers wish to gather evidence to support.

The Null Hypothesis, Ho, is usually the hypothesis for which the researchers wish to gather evidence to reject.

 The Null Hypothesis, Ho, is always expressed in the form of an equality.

i.e.  
Ho: m = 5.8 lbs.
or
Ho: p= 3.2%

example: The Georgia Department of Transportation claims the average number of accidents on I-285 each day is 3.2. We believe the claimed average is too small. State Ho and Ha.

 

Ho: m = 3.2

Ha: m > 3.2

example: Kennesaw State University claims the average GPA is 2.73. We believe the claimed average is incorrect. State Ho and Ha.

 

Ho: m = 2.73

Ha: m 2.73

A one-tailed test is one where Ha is directional and includes < or >.

or

A two-tailed test is one where Ha no direction is indicated and utilizes .

 

What can go right?

 

Decide Ho is true when it is true.
Decide Ha is true when it is true.

 

What can go wrong?

 

Decide Ho is true when it is false. Type II error

Decide Ha is true when it is false. Type I error

The level of significance a, is the probability of making a Type I error.

The rejection region is the set of possible values for which the null hypothesis will be rejected. This region will depend on a.

In specifying the rejection region for a hypothesis, the value at the boundary of the rejection region is called the critical value.

What is the critical value for a two-tailed test and a 95% level of confidence?

The critical values for a two-tailed test are the same as .

This is not true for one-tailed tests!

What is the critical value for a one-tailed test and a 95% level of confidence?

Level of confidence

1-a

a Two-tailed critical value One-tailed critical value
90% 10% 1.645  
95% 5% 1.96 1.645
98% 2% 2.33  
99% 1% 2.575  

 

 

 

To determine if the Null Hypothesis should be rejected in favor of the Alternative, a test statistic is computed. The test statistic is computed as a z-score using the Null Hypothesis as the population average and a sample information for the observation and estimated standard deviation.

example: Consider the following:

Ho: m = 58

Ha: m > 58

A sample of 49 observations yields an average of 63 and standard deviation of 8. At a confidence level of 95%, is this enough evidence to reject Ho in favor of Ha or is this just chance variation?

 

This is a one-tailed test and the critical value is 1.645. The test statistic is

.

 

The test statistic, z = 4.38 falls into the rejection region. Thus, we reject the Null Hypothesis in favor of the Alternative Hypothesis.

 

example: Consider the following:

Ho: m = 50

Ha: m 50

A sample of 100 observations yields an average of 51.5 and standard deviation of 10. At a confidence level of 95%, is this enough evidence to reject Ho in favor of Ha or is this just chance variation?

This is a two-tailed test and the critical value is z = 1.96. The test statistic is

.

The test statistic, z =1.5 does not fall into the rejection region. Thus, we do not reject the Null Hypothesis in favor of the Alternative Hypothesis.

 

 

 

example: The airline industry claims that the average age of a jet is 10 years with a standard deviation of 7.5 years. One critic of the Fly by Night airline claims that their jets are too old. A sample of 40 Fly by Night jets yields an average age of 20 years. Do you agree with the critic and Fly by Night's jets are too old or is this just chance variation? Test at 99% confidence.

 

 

Syllabus Files ] Tips for Success ] Intro ] Chapter 2: Numerical Methods ] Chapter 3: Probability ] Chapter 4: Probability Distributions ] Chapter 5: The Normal Curve ] Chapter 5: Central Limit Theorem ] Chapter 4: The Binomial Probability Distribution ] Questionnaire ] Chapter 1 ] Chapter 6: Confidence Intervals ] Chapter 6: Required Sample Size ] Chapter 6: Estimation of Proportion ] Required Sample Size (Updated) ] [ Chapter 7: Hypothesis Testing ] Chapter 7: Applications of Hypothesis Testing ] Sample Tests ]