Chapter 10.2, Problem 10.57E

To determine

To find: the conclusion and give appropriate evidence to support the conclusion.

Answer to Problem 10.57E

There is enough evidence to help the claim that the mean study hours per night are less than 2.5 hours.

Explanation of Solution

Given:

  $\begin{array}{c}n=269\\ \overline{x}=137\\ s=65\end{array}$

Claim: Mean is less than 150 minutes (2.5 hours)

Formula used:

  $t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$

Calculation:

Assume

  $\alpha =0.05$

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis statement is that the population mean is equal to the value given in the claim. If the null hypothesis is the claim, then the alternative hypothesis statement is the opposite of the null hypothesis.

  $\begin{array}{c}{H}_0:\mu =150\\ H_1:\mu <150\end{array}$

The value of the test statistic is

  $t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}=\frac{137-150}{65/\sqrt{269}}=-3.28$

The P-value is the probability of getting a value more extreme or equal to the standardized test statistic $z$ , suppose that the null hypothesis is true

  $P=P(z<-3.28)=0.0005$

If the P-value is lesser than the significance level $\alpha$ , then the alternative hypothesis is accepted and null hypothesis is rejected.

  $P<0.05\Rightarrow {\text{Reject H}}_0$

There is enough evidence to help the claim that the mean study hours per night is less than 2.5 hours.