More Hypothesis Testing Chapter 9, part D More Hypothesis Testing

V. Tests about : Small sample case If you have a small (n<30) sample, it’s appropriate to use the t-distribution for hypothesis testing. Your test statistic, has a t-distribution with (n-1) degrees of freedom.

A. One-Tailed Test Historically the population of Hanover students has a cumulative G.P.A. of no more than 2.77. Suppose you want to test the hypothesis that (due to smarter students) the mean GPA is higher. Ho:   2.77 Ha:  > 2.77

The Test In a sample of 25 students, you calculate a mean GPA of 2.8491 and a standard deviation of .4796. Test the hypothesis at the 95% confidence. Your critical t.05 with 24 degrees of freedom is 1.711. The test statistic: =.8246 You can’t reject Ho and must conclude that the GPA has not increased.

B. Two-Tailed Test A golf ball producer tests driving distance of a new golf ball. Due to PGA specifications, the distance needs to average 280 yards. Ho:  = 280 yds. Ha:   280 yds.

The Test You sample n=10 golf balls, calculate a sample mean of 282.2 yards and a standard deviation of 13.53. With 9 degrees of freedom, the critical t.025=±2.262 Your test statistic is .514, so you cannot reject Ho. Your conclusion is that the golf balls are o.k. Can you do this?

VI. Tests about a population proportion The techniques are virtually identical to those testing a population mean. “p” is the population proportion, and p0 is the hypothesized value of the population proportion. One and two-tailed tests are structured in the same way as before.

A. Testing a proportion hypothesis The main difference when dealing with a proportion problem is in the calculation of the standard error. Recall from chapter 7: We also have to test to see if we can safely apply the Central Limit Theorem. If np>=5 and n(1-p)>=5, our sample is “large enough”.

B. An example A restaurant believes that 30% of their customers do not drink water with their meal. In a recent sample of 480 customers, 128 did not drink their water. Use this sample to test their belief at a 95% confidence level.

Ho:  = .30. Ha:   .30 p-bar = 128/480 =.2667. = .0209 The critical value is  1.96 and the test statistic is Z = -1.5933 so we fail to reject the null.