Question

A college professor suspects that students at his school are getting less than 8 hours of sleep a night, on average. To test his belief, the professor asks a random sample of $28$ students, “How much sleep did you get last night?” Here are the data (in hours): $96868866.56794345611636610784.5977$

Do these data provide convincing evidence in support of the professor’s suspicion? Carry out a significance test at the $a=0.05$level to help answer this question.

Short answer

Yes, there is sufficient evidence to support the suspicion of professor.

Step-by-step solution

Given information

The data set is:$96868866.56794345611636610784.5977$

Concept

The test statistic is computed as: $t=\frac{{\displaystyle \frac{\overline{x}-\mu }{s}}}{\sqrt{n}}$

$\overline{x}=$Sample mean

$\mu =$Population mean

$n=$ Sample size

$s=$ Sample standard deviation

Calculation

Here,

be the average sleeping hours.

The null and alternative hypotheses are:$$\begin{gathered} H_0:\mu =8 \\ {H}_a:\mu <8 \end{gathered}$$

The obtained excel output is:

Here, the p-value is $0.0006$ Here, is the p-value.

Thus, at a $5\%$ significance level, there is sufficient evidence to support the suspicion of the professor.