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 25 scores from all of the students who took the test. She sees that the mean score is 139 with a standard deviation of 16.5865. The superintendant wants to know if the standard deviation has changed this year. Previously, the population standard deviation was 29. Is there evidence that the standard deviation of test scores has decreased at the α=0.025

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. Reject or fail to reject

Answer 1

Solution :

Step 1 : Null and alternative hypotheses :

The null and alternative hypotheses would be as follows :

​​​​​

Step 2 : Critical value

Significance level = 0.025

Degrees of freedom = (n - 1) = (25 - 1) = 24

Since, our test is left-tailed test, therefore we shall obtain the left-tailed critical value of chi-square at 0.025, which is given as follows :

Step 3 : Test statisic

To test the hypothesis we shall use chi-square test for variance. The test statistic is given as follows :

Where, n is sample size, s is sample standard deviation and σ is hypothesized value of population standard deviation under H_{​​​​​​0}.

We have, n = 25, s = 16.5865 and σ = 29

The value of the test statistic is 7.851.

Step 4 : Decision

For left-tailed test, we make decision rule as follows :

If value of the test statistic is less than the left-tailed critical value, then we reject the null hypothesis. Otherwise we fail to reject the null hypothesis.

Test statistic value = 7.851

Critical value = 12.401

(7.851 < 12.401)

Since, value of the test statistic is less than the left-tailed critical value, therefore we shall reject the null hypothesis at 0.025 significance level.

Step 5 : Conclusion

At 0.025 significance level, there is sufficient evidence to conclude that the standard deviation of test scores has decreased.

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