Question

If the quality of teaching is similar in a school, the scores on a standardized test will have a standard deviation of 19. The superintendent wants to know if there is a disparity in teaching quality, and decides to investigate whether the standard deviation of test scores has changed. She samples 22 random students and finds a mean score of 110 with a standard deviation of 16.709. Is there evidence that the standard deviation of test scores has decreased at the α=0.05 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

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: σ = 19
Alternative Hypothesis, Ha: σ < 19

Rejection Region
This is left tailed test, for α = 0.05 and df = 21
Critical value of Χ^2 is 11.591
Hence reject H0 if Χ^2 < 11.591

Test statistic,
Χ^2 = (n-1)*s^2/σ^2
Χ^2 = (22 - 1)*16.709^2/19^2
Χ^2 = 16.241

As the test statistic lies in the rejection region, reject the null hypothesis.

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