Q 2.1. - The Practice of Statistics for AP

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): $9 6 8 6 8 8 6 6.5 6 7 9 4 3 4 5 6 11 6 3 6 6 10 7 8 4.5 9 7 7$

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.

Step-by-Step Solution

Answer

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

Step 1: Given information

The data set is: $9 6 8 6 8 8 6 6.5 6 7 9 4 3 4 5 6 11 6 3 6 6 10 7 8 4.5 9 7 7$

Step 2: 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

Step 3: 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.