Question

Toastmasters International cites a report by Gallop Poll that $40\%$ of Americans fear public speaking. A student believes that less than $40\%$ of students at her school fear public speaking. She randomly surveys$361$ schoolmates and finds that$135$ report they fear public speaking. Conduct a hypothesis test to determine if the percent at her school is less than $40\%.$

Short answer

They lack sufficient information to claim that less than $40\%$of pupils at her school are afraid of public speaking.

Step-by-step solution

Given information

Sample size, $n=361$

According to the report by Gallop Poll, the proportion of Americans who fear public speaking$=40\%$

Explanation

The null hypothesis states that $40\%$of Americans fear public speaking and the alternative hypothesis states that less than $40\%$of Americans fear public speaking.

$$\begin{gathered} H_0:p=0.40 \\ {H}_0:p<0.40 \end{gathered}$$

The Z test statistic is:

$$\begin{gathered} z=\frac{p^{\wedge }-p}{\sqrt{{\displaystyle \frac{p\left(1-p\right)}{n}}}} \\ {p}^{\wedge }=\frac{x}{n} \\ =\frac{135}{361} \\ =0.374 \end{gathered}$$

$$\begin{gathered} z=\frac{p^{\wedge }-p}{\sqrt{{\displaystyle \frac{p\left(1-p\right)}{n}}}} \\ =\frac{0.374-0.40}{\sqrt{{\displaystyle \frac{p\left(1-p\right)}{n}}}} \\ =\frac{-0.026}{0.0257} \\ =-1.01 \end{gathered}$$