Question

People in a random sample of 237 students enrolled at a liberal arts college were asked questions about how many hours of sleep they get each night. The sample mean sleep duration (average hours of daily sleep) was 7.72 hours and the sample standard deviation was 1.02 hours. The recommended number of hours of sleep for college-age students is 8.4 hours. Is there convincing evidence that the population mean sleep duration for students at this college is less than the recommended number of 8.4 hours

Answer 1

There is enough evidence to support the claim that the population mean of the students at this college is less than the recommended number of 8.4 hours.

Explanation:

We are given the following in the question:

Population mean, μ = 8.4 hours

Sample mean,
`\bar{x}` = 7.72 hours

Sample size, n = 237

Alpha, α = 0.01

Sample standard deviation, s = 1.02 hours

First, we design the null and the alternate hypothesis

`H_(0): \mu = 8.4\text{ hours}\\H_A: \mu < 8.4\text{ hours}}`

We use one-tailed t test to perform this hypothesis.

Formula:

`t_(stat) = \displaystyle\frac{\bar{x} - \mu}{(s)/(√(n)) }`

Putting all the values, we have

`t_(stat) = \displaystyle(7.72 - 8.4)/((1.02)/(√(237)) ) = -10.2632`

Now,
`t_(critical) \text{ at 0.05 level of significance, 236 degree of freedom } = -2.3422`

Since, the calculated test statistic is less than the critical value, we fail to accept the null hypothesis and reject it. We accept the alternate hypothesis.

Conclusion:

Thus, there is enough evidence to support the claim that the population mean of the students at this college is less than the recommended number of 8.4 hours.