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Question

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population. Tests in the author's statistics classes have scores with a standard deviation equal to 14.1. One of his last classes had 27 test scores with a standard deviation of 9.3. Use a 0.01 significance level to test the claim that this class has less variation than other past classes. Does a lower standard deviation suggest that this last class is doing better?

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Given (in the output):

s=9.3s=9.3

n=27n=27

α=0.01\alpha=0.01

Claim: Standard deviation is 14.1

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis needs to contain an equality and the value mentioned in the claim. If the claim is the null hypothesis, then the alternative hypothesis states the opposite of each other.

H0:σ=14.1H_0:\sigma=14.1

H1:σ14.1H_1:\sigma\neq 14.1

Compute the value of the test statistic:

χ2=n1σ02s2=27114.129.3211.3110\chi^2=\dfrac{n-1}{\sigma^2_0}s^2=\dfrac{27-1}{14.1^2}\cdot 9.3^2\approx 11.3110

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, assuming that the null hypothesis is true. The P-value is the number (or interval) in the column title of the chi-square distribution containing the χ2\chi^2-value in the row df=n1=271=26df=n-1=27-1=26:

0.01=2×(10.995)<P<2×(10.99)=0.020.01=2\times (1-0.995)<P<2\times (1-0.99)=0.02

If the P-value is less than the significance level α\alpha, then reject the null hypothesis.

P>0.01 Fail to reject H0P>0.01\Rightarrow \text{ Fail to reject } H_0

There is not sufficient evidence to reject the claim that the scores have a standard deviation equal to 14.1 and thus the last class does not appear to be doing better.

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