Question

1. During recent seasons, Major League Baseball has been criticized for the length of the games. A report indicated that the average game lasts 3 hours and 30 minutes. A random sample of 17 games was checked for the times to completion. The mean game time for this sample was 2.955294 hours and the sample standard deviation was 0.13571 hours. Assume that the game time follows a normal distribution.

Can we conclude that the mean time for a game is less than 3.5 hours? Use the .05 significance level.

Answer 1

The provided sample mean is and the sample standard deviation is s=0.13571, and the sample size is n=17.

(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 t-test for one mean, with unknown population standard deviation, will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is tc​=−1.746.

The rejection region for this left-tailed test is R=t:t<−1.746

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that t=−16.549<tc​=−1.746, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0, and since p=0<0.05, 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 3.5, at the 0.05 significance level.

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