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 teen girls smoke to stay thin? The alternative hypothesis is:

$$\begin{gathered} a.p<0.30 \\ b.p\le 0.30 \\ c.p\ge 0.30 \\ d.p>0.30 \end{gathered}$$

Short answer

The correct option is d.$H_a:$$p>0.30$.

Step-by-step solution

Introduction

The null hypothesis of a statistical hypothesis test always predicts that there is no impact or association between variables.

Explanation Part a

From the given information it is said that,

$x=63$

$n=273$

When finding the null hypothesis i.e

$$\begin{gathered} p=\frac{x}{n} \\ p=\frac{63}{273} \\ p=23 \end{gathered}$$

We can see that value of $p$is less than $30\%$, Therefore, it is not the correct option.

Explanation Part b

From the given information it is said that,

$x=63$is the number of teen girls who smoke to stay thin.

$n=273$is the group of teen girls randomly selected.

When finding the null hypothesis i.e

$$\begin{gathered} p=\frac{x}{n} \\ p=\frac{63}{273} \\ p=23 \end{gathered}$$

Therefore,

$p\le 0.30$cannot be the correct answer as the number of girls smoking to stay thin is less than$30\%$and here it is less than and equal to$30\%$

Explanation Part c

From the given information it is said that,

$x=63$is the number of teen girls who smoke to stay thin.

$n=273$is the group of teen girls randomly selected.

When finding the null hypothesis i.e

$$\begin{gathered} p=\frac{x}{n} \\ p=\frac{63}{273} \\ p=23 \end{gathered}$$

Therefore,

$p\ge 0.30$cannot be the correct answer as the number of girls smoking to stay thin is less than$30\%$and here it is greater than and equal to$30\%$

Explanation Part d

As the null hypothesis states that either $30$or less than $30\%$of girls smoke to stay thin, hence, the alternate hypothesis has to say that greater than $30\%$of girls smoke to stay thin. The alternate hypothesis always opposes the null hypothesis.

To calculate we are given,

$x=63$is the number of teen girls who smoke to stay thin.

$n=273$is the group of teen girls randomly selected.

$$\begin{gathered} p=\frac{x}{n} \\ p=\frac{63}{273} \\ p=23 \end{gathered}$$

$23\%$of the girls actually smoked to stay thin. The number of girls smoking to stay thin is less than $30\%.$