Question

Previously, an organization reported that teenagers spent $4.5$hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was $4.75$hours with a sample standard deviation of $2.0$. Conduct a hypothesis test, the Type I error is:

a. to conclude that the current mean hours per week is higher than $4.5$, when in fact, it is higher

b. to conclude that the current mean hours per week is higher than $4.5$, when in fact, it is the same

c. to conclude that the mean hours per week currently is $4.5$, when in fact, it is higher

d. to conclude that the mean hours per week currently is no higher than $4.5$, when in fact, it is not higher

Short answer

The answer is option b.

which says that the current mean hours per week is higher than $4.5$, when in fact, it is the same.

Step-by-step solution

Given Information

$$\begin{gathered} n=15 \\ \overline{x}=4.75 \\ \sigma =2 \\ Wecanwritefromtheproblemthat: \\ {H}_0:\mu =4.5;H_a:\mu >4.5 \end{gathered}$$

The current mean hours per week is higher than $4.5$is a statement of inequality. Therefore, it is the alternative hypothesis.

Hence, it follows that the null hypothesis is the current mean hours per week is $4.5$.

Explanation

Null Hypothesis

In a hypothesis test, null and alternative hypotheses are identified from the original claim. The null hypothesis $H_0$is a statement of equality of a population parameter to a certain value.

By definition of Type I error, the Type I error in the task is:

To conclude that the current mean hours per week is higher than $4.5$, when in fact, it is the same.

Therefore, the correct answer is option b.