A survey of 162 students...

Question

A survey of 162 students nationwide showed a mean ACT score of 77.6. A survey of 138 Ohio scores showed a mean of 83.9. If the population standard deviation in each case is 16.62, and at alpha 0.05, test the claim that the nationwide score is less than the Ohio score. Compute for the critical value.

Answer 1

Hello Faith, happy to help!

This question is a hypothesis test for two different population means. In other words, we need to take both population parameters into consideration as we go through the problem. First step, let's set up our Alternative Hypothesis.

Ha: mu (nationwide scores) - mu (Ohio scores) < 0

where mu is the population mean.

Since the sign of the alternative hypothesis is a "less than" sign, we are using a left-tailed one-sided hypothesis test. The critical value can be found using the z table. If our alpha is 0.05, the corresponding z value is roughly -1.64.

In order to complete our hypothesis test, we need to compare our critical value to the z test statistic. In a two population hypothesis test, the z test statistic equals the following:

Z Test Statistic = (Xbar1 - Xbar2 - D0) / ( Square root of ((standard deviation1^2 / sample size 1) + (standard deviation2^2 / sample size2))

Z Test Statistic = (77.6 - 83.0 - 0) / (Square root of ((16.62^2 / 162) + (16.62^2 / 138))

Z Test Statistic = -6.3 / (Square root of (3.7067)) = -6.3 / 1.92528 = -3.27.

Upon comparing the critical value (-1.64) to the z test statistic (-3.27), we have strong evidence to support our alternative hypothesis that the nationwide score is less than the Ohio score with a p value of 0.00054 (found by looking up the percentage of a z score of -3.27 on the.z table).

Again, I hope this helps. If you would like to meet and discuss, please feel free to schedule time to meet with me on Wyzant :)