Question

Sleep habits of new Yorkers. New York is known as the city that never sleeps. A random sample of 25 New Yorkers was asked how much sleep they get per night. The statistical summaries of these data are shown below. Do these data provide strong evidence that New Yorkers sleep more of less than 8 hours a night on average?

Write the hypotheses in symbols and in words

check the conditions, then calculate the test statistic, T, and the associated degrees of freedom

what is the conclusion of the hypothesis test

if you were to construct a 95% confidence interval that corresponded to this hypothesis test, would you expect 8 hours in the interval?

n    x s min    max
25 7.73 0.77    6.17 9.78

Answer 1

We will consider the level of significance to be 5%

The provided sample mean is and the sample standard deviation is s=0.77, and the sample size is n=25.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: i.e. new yorkers sleep 8 hours a day on average

Ha: i.e. new yorkers do not sleep 8 hours a day on average

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=0.05, and the critical value for a two-tailed test is tc​=2.064

The rejection region for this two-tailed test is R={t:∣t∣>2.064}

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that , it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0923, and since p=0.0923≥0.05, 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 8, at the 0.05 significance level.

Confidence Interval

The 95% confidence interval is 7.412<μ<8.048.

Yes, we would expect 8 hours in the interval since we have failed to reject the null hypothesis.

Graphically

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