Question

"The Problem with Angels," by Cyndy Dowling although this problem is wholly mine, the catalyst came from the magazine, Time. On the magazine cover I did find the realm of angels tickling my mind. Inside, $69\%$I found to be in angels, Americans do believe.

Then, it was time to rise to the task, ninety-five high school and college students I did ask. Viewing all as one group, random sampling to get the scoop. So, I asked each to be true, "Do you believe in angels?" Tell me, do! Hypothesizing at the start, totally believing in my heart that the proportion who said yes would be equal on this test. Lo and behold, seventy-three did arrive, out of the sample of ninety-five. Now your job has just begun, Solve this problem and have some fun.

Short answer

The null hypothesis is not rejected and the proportion of people who believe in angels is $69\%$.

Step-by-step solution

Given Information

The phrases from "The problem with Angels" is given. It is mentioned that $69\%$of Americans believe in Angels.

Explanation

Hypothesis:

According to the null hypothesis, there is$69\%$of people who believe in angels.

$$\begin{gathered} H_0:p=0.69 \\ {H}_0:p\ne 0.69 \end{gathered}$$

The normal distribution is,

$N\left(0.69,\sqrt{\frac{\left(0.69\right)\left(0.31\right)}{95}}\right)$

Calculate localid="1649763939968" $p^$

localid="1649763934474" $$\begin{gathered} P^{\wedge }=\frac{x}{n} \\ \text{wherenissamplesize} \\ {P}^{\wedge }=\frac{73}{95} \\ {P}^{\wedge }=0.77 \end{gathered}$$

Conclusion

The Z test statistic is,

$$\begin{gathered} z=\frac{p^{\wedge }-p}{\sqrt{{\displaystyle \frac{p\left(1-p\right)}{n}}}} \\ =\frac{0.77-0.69}{\sqrt{{\displaystyle \frac{p\left(1-p\right)}{n}}}} \\ =\frac{0.08}{0.0475} \\ =1.686 \end{gathered}$$