Question

Over the past few decades, public health officials have examined the link between weight concerns and teen girls' smoking. Researchers surveyed a group of $273$randomly selected teen girls living in Massachusetts (between $12$and $15$years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin?

After conducting the test, your decision and conclusion are

a. Reject$H_0$: There is sufficient evidence to conclude that more than$30\%$of teen girls smoke to stay thin.

b. Do not reject$H_0$: There is not sufficient evidence to conclude that less than $30\%$of teen girls smoke to stay thin.

c. Do not reject $H_0$: There is not sufficient evidence to conclude that more than $30\%$ of teen girls smoke to stay thin.

d. Reject $H_0$: There is sufficient evidence to conclude that less than $30\%$ of teen girls smoke to stay thin.

Short answer

c. Do not reject $H_0$: There is not sufficient evidence to conclude that more than$30\%$of teen girls smoke to stay thin.

Step-by-step solution

Given information

$x=63,n=273,p_0=0.30,a=0.05$

Explanation

State the hypothesis:

The null hypothesis states that less than or equal to$30\%$of teen girls smoke to stay thin. In symbols:

$H_0:p\le 0.30$

The alternative hypothesis states that more than $30\%$of teen girls smoke to stay thin. In symbols:

$H_a:p>0.30$

From the z table, the critical value of z value for the critical region is:

$$\begin{gathered} \stackrel{⏜}{P}=\frac{x}{n} \\ =\frac{63}{273} \\ =0.231 \end{gathered}$$

The test statistic is:

$$\begin{gathered} z=\frac{\stackrel{⏜}{p}-p_0}{\sqrt{{\displaystyle \frac{p_0\left(1-p_0\right)}{n}}}} \\ =\frac{0.231-0.30}{\sqrt{{\displaystyle \frac{0.30\left(1-0.30\right)}{273}}}} \\ =\frac{-0.069}{0.027735} \\ =-2.50 \end{gathered}$$

Since the null hypothesis is not rejected, there is not sufficient evidence to conclude that more than $30\%$of teen girls smoke to stay thin.