Question

In a school district, all sixth grade students take the same standardized test. The superintendant of the school district takes a random sample of 26 scores from all of the students who took the test. She sees that the mean score is 101 with a standard deviation of 10.1793. The superintendant wants to know if the standard deviation has changed this year. Previously, the population standard deviation was 21. Is there evidence that the standard deviation of test scores has decreased at the α=0.01 level? Assume the population is normally distributed. Step 1 of 5: State the null and alternative hypotheses. Round to four decimal places when necessary.

Step 2 of 5:

Determine the critical value(s) of the test statistic. If the test is two-tailed, separate the values with a comma. Round your answer to three decimal places.

Step 3 of 5:

Determine the value of the test statistic. Round your answer to three decimal places.

Step 4 of 5:

Make the decision.

Step 5 of 5:

What is the conclusion?

Answer 1

Solution:

Given:

Sample size = n = 26

Sample mean = 101

Sample standard deviation = 10.1793

The population standard deviation = 21

We have to test if  there is evidence that the standard deviation of test scores has decreased at the α=0.01 level.

Step 1 of 5: State the null and alternative hypotheses.

Vs

( Left tailed test)

Step 2 of 5: Determine the critical value(s) of the test statistic.

df = n - 1 = 26 -1 = 25

α=0.01

Since this is left tailed test, find Area = 1 - α= 1 - 0.01 = 0.99

the critical value = 11.524

Step 3 of 5: Determine the value of the test statistic.

Step 4 of 5: Make the decision.

Decision Rule:

Reject null hypothesis H0, if Chi square test statistic < Chi-square critical value = 11.524, otherwise we fail to reject H0.

Since Chi square test statistic = < Chi-square critical value = 11.524, we reject null hypothesis H0.

Step 5 of 5: What is the conclusion?

There is sufficient evidence that the standard deviation of test scores has decreased.