Question

In addition to its core business of bagels and coffee, Brueggers Bagles also sells hot chocolate for the noncoffee crowd. Customer research shows that the ideal temperature for hot chocolate is 142 F (“hot” but not “too hot”). A random sample of 24 cups of hot chocolate is taken at various times, and the temperature of each cup is measured using an ordinary kitchen thermometer that is accurate to the nearest whole degree. The sample mean is 141.375 with a sample standard deviation of 1.99592. At a=.10, does this sample evidence show that the true mean differs from 142?

140 140 141 145 143 144 142 140

145 143 140 140 141 141 137 142

143 141 142 142 143 141 138 139

Answer 1

The provided sample mean is and the sample standard deviation is , and the sample size is n=24.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ = 142

Ha: μ ≠ 142

This corresponds to a two-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=0.10, and the critical value for a two-tailed test is tc​=1.714.

The rejection region for this two-tailed test is R={t:∣t∣>1.714}

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣t∣=1.534≤tc​=1.714, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.1387, and since p=0.1387≥0.10, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ is different than 142, at the 0.10 significance level.

Graphically

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