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.

At a significance level of $a=0.05$, what is the correct conclusion?

a. There is enough evidence to conclude that the mean number of hours is more than$4.75$

b. There is enough evidence to conclude that the mean number of hours is more than 4.5

c. There is not enough evidence to conclude that the mean number of hours is more than 4.5

d. There is not enough evidence to conclude that the mean number of hours is more than 4.75

Instructions: For the following ten exercises,

Hypothesis testing: For the following ten exercises, answer each question.

a. State the null and alternate hypothesis.

b. State the p-value.

c. State alpha.

d. What is your decision?

e. Write a conclusion.

f. Answer any other questions asked in the problem

Short answer

The correct conclusion is option (c). There is not enough evidence to conclude that the mean number of hours is more than $4.5.$

Step-by-step solution

Given information

$M=4.75,S=2.00,n=15,a=0.05$

Explanation

Hypothesis: The null hypothesis suggests that teenagers spend less than or equal to $4.5$hours per week on their phones. According to the alternative theory, teenagers spend more than $4.5$hours every day on their phones. The hypothesis is represented as:

$H_0:\mu \le 4.5$

$H_a:\mu >4.5$

The degree of freedom is:

$$\begin{gathered} df=n-1 \\ =15-1 \\ =14 \end{gathered}$$

From that t distribution table, for a right-tailed test with $a=0.05$for $df=14$, the critical value of t values for the critical region are $t=1.761$.

The test statistic is:

localid="1649832749610" $$\begin{gathered} t=\frac{M-\mu }{{\displaystyle \frac{S}{\sqrt{n}}}} \\ =\frac{4.75-4.5}{{\displaystyle \frac{2}{\sqrt{15}}}} \\ =\frac{0.25}{0.5164} \\ =0.48 \end{gathered}$$

Since the test statistic does not fall within the critical zone null hypothesis is not rejected. Therefore we can't say that the average number of hours spent on the phone by teenagers is higher than$4.5$.