Question

1. A researcher claims that the average cost of men’s athletic shoes is less than $80. He selects a random sample of 36 pairs of shoes from a catalog and finds that the average cost is $75. Is there enough evidence to support the researcher’s claim at Alpha = 0.10 . Assume the population standard deviation is 19.2.

Define

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Answer 1

The provided sample mean is and the known population standard deviation is , and the sample size is

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ=80

Ha: μ<80

This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.10, and the critical value for a left-tailed test is zc​=−1.28.

The rejection region for this left-tailed test is R={z:z<−1.28}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z=−1.563<zc​=−1.28, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0591, and since p=0.0591<0.10, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 80, at the 0.10 significance level.

Graphically

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