Question

A statistics instructor believes that fewer than $20\%$ of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys $84$of her students and finds that$11$of them attended the midnight showing.

At a$1\%$ level of significance, an appropriate conclusion is:

a. There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than$20\%$.

b. There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is more than $20\%.$

c. There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than$20\%$.

d. There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is at least $20\%$

Short answer

The appropriate conclusion is option (a) . There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than $20\%$.

Step-by-step solution

Given information

$x=11,n=84,p_0=0.20,a=0.01$

Explanation

Hypothesis: The null hypothesis claims that more than or equal to $20\%$of Evergreen Valley College students saw the latest Harry Potter film at midnight on its opening night. According to the alternative hypothesis, fewer than $20\%$of Evergreen Valley College students attended the midnight premiere of the latest Harry Potter film. The hypothesis can be represented as

$H_0:p\ge 0.20$

$H_a:p<20$

From the Z table, the critical value z is $z=-2.33ata=0.01$

The sample proportion is:

$$\begin{gathered} \stackrel{⏜}{P}=\frac{x}{n} \\ =\frac{11}{84} \\ =0.131 \end{gathered}$$

The test statistic is:

$$\begin{gathered} z=\frac{\stackrel{⏜}{p}-p_0}{\sqrt{{\displaystyle \frac{p_0\left(1-p_0\right)}{n}}}} \\ =\frac{0.131-0.20}{\sqrt{{\displaystyle \frac{0.20\left(1-0.20\right)}{84}}}} \\ =\frac{-0.069}{0.04364} \\ =-1.58 \end{gathered}$$

There is insufficient information to infer that fewer than $20\%$of Evergreen Valley College students attended the midnight premiere of the latest Harry Potter film because the null hypothesis was not rejected.