Question

A standardized test is given to a sixth grade class. Historically the mean score has been 126 with a variance of 12. The superintendent believes that the variance of performance may have recently decreased. She randomly sampled 23 students and found a mean of 116 with a standard deviation of 2.0677. Is there evidence that the standard deviation has decreased at the a=.05 level?

Step 1 of 5: State the hypotheses in terms of the standard deviation. Round the standard deviation 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, seperate the values with a comma. Round your answer to three decimal places.

Step 3 of 5: Determine value of the test statistic. Round your answer to three decimal places.

Step 4 of 5: Make the decision. Reject Null or Fail to Reject

Step 5 of 5: What is the conclusion? Sufficient evidence or not sufficient evidence

Answer 1

The provided sample variance is and the sample size is given by n=23.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:

Ha:

This corresponds to a left-tailed test, for which a Chi-Square test for one population variance will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the rejection region for this left-tailed test is

(3) Test Statistics

The Chi-Squared statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that , it is then concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is sufficient evidence to claim that the population variance is less than 12, at the 0.05 significance level.

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