Question

Getting Enough Sleep? It is generally recommended that adults sleep at least 8 hours each night. One of the authors recently asked some of her students (undergraduate and graduate students at Harvard) how many hours each had slept the previous night, curious as to whether her students are getting enough sleep. The data are displayed in Figure 6.13 . The 12 students sampled averaged 6.2 hours of sleep with a standard deviation of 1.70 hours, Assuming this sample is representative of all her students, and assuming students need at least 8 hours of sleep a night, does this provide evidence that, on average, her students are not getting enough sleep?

Short answer

To conclude whether the students are getting enough sleep, one needs to calculate the test statistic using the given sample mean, standard deviation, and sample size and compare this calculated t-value to the critical t-value for a specified confidence level. Decision will be made based on this comparison.

Step-by-step solution

Define the Hypotheses

The null hypothesis \(H_0\) is that the average student gets enough sleep, expressed as \(H_0: \mu = 8\). The alternative hypothesis \(H_a\), which we are testing, is that the average student does not get enough sleep, expressed as \(H_a: \mu < 8\) . Here, \( \mu \) represents the population mean.

Compute the Test Statistic

The test statistic for a one-sample mean test is a t-score, calculated using the formula \(t = \frac{ \overline{X} - \mu_{H_0}}{s / \sqrt{n}}\). Here, \(\overline{X} = 6.2\) is the sample mean, \( \mu_{H_0} = 8\) is the mean under the null hypothesis, \(s = 1.70\) is the sample standard deviation, and \(n = 12\) is the sample size. Plugging these values into the formula, we get \(t = \frac{6.2 - 8}{1.70 / \sqrt{12}}\).

Find the Critical Value

The critical value corresponds to a significance level (or alpha) of 0.05 with a sample size of 11 degrees of freedom (since df = n - 1). You can find this value on the t-distribution table or using a t-distribution calculator online. This will be the boundary which will determine whether we reject \(H_0\) or fail to reject \(H_0\).

Make the Decision

If the calculated test statistic is less than the critical value, then we reject the null hypothesis, confirming the alternative hypothesis. If not, we fail to reject the null hypothesis, meaning we do not have enough evidence to suggest the students are not getting enough sleep.