Question

Government Surveillance. Gallup conducted a national survey of $1008$American adults, asking "As you may know, as part of its efforts to investigate terrorism, a federal government agency obtained records from larger U.S. telephone and Internet companies in order to compile telephone call logs and Internet communications. Based on what you have heard or read about the program, would you say that you approve or disapprove of this government program?" Of those surveyed, $534$said they disapprove.

a. Determine and interpret the sample proportion.

b. At the $5\%$significance level, do the data provide sufficient evidence to conclude that a majority (more than $50\%$) of American adults disapprove of this government surveillance program?

Short answer

(a) The sample proportion is $0.5298$.

(b) The findings give sufficient evidence to conclude that the majority of adults in the United States oppose the government's initiative.

Step-by-step solution

Part (a) Step 1: Given information

The significance level is $5\%$

$x=534$

$n=1008$

Part (a) Step 2: Explanation

The sample proportion is expressed as

$\hat{p}=\frac{x}{n}$

The sample proportion is

$\hat{p}=\frac{534}{1008}$

$=0.5298$

As a result, the sample proportion is $0.5298$, indicating that of those polled oppose the government policy.

Part (b) Step 1: Given information

The significance level is $5\%$

$x=534$

$n=1008$

Part (b) Step 2: Explanation

Lets calculate the value of$n{p}_0$

$n{p}_0=\left(1008\right)(0.5)$

$=504$

Calculate the value of$n\left(1-p_0\right)$

$n\left(1-p_0\right)=\left(1008\right)(1-0.5)$

$=1008(0.5)$

$=504$

Both are greater than $5$. So one proportion $z-$test is appropriate to use.

The null hypothesis is

$H_0:p_0=0.5$

The alternate hypothesis is

$H_0:p_0>0.5$

The expression for $z$is

$z=\frac{\hat{p}-p_0}{\sqrt{\frac{P_0\left(1-p_0\right)}{n}}}$

The $z$value is calculated as

$z=\frac{0.5298-0.5}{\sqrt{\frac{0.5(1-0.5)}{1008}}}$

$=\frac{0.0298}{0.0157}$

$=1.89$

And the $\alpha =0.05$, the critical value from the standard table is

$z_{0.05}=1.645$

Because the test is a right-tailed test, a positive value is used.

The test statistic is in the negative range. As a result, the hypothesis $H_0$is ruled out.

As a result, the findings give sufficient evidence to conclude that the majority of adults in the United States oppose the government's initiative.