Chapter 16, Problem 1E

a.

To determine

Explain the meaning of $\widehat{p}=0.49$ in this situation.

Explanation of Solution

In Pew research, the random sample of 799 U.S. teens about the internet usage reports that 49% teens misrepresented their age online to access websites and online services.

Let p be the population proportion of the U.S teens who misrepresented their age and $\widehat{p}$ acts as the best estimate of p.

Here, $\widehat{p}$ be the sample proportion and it denotes the proportion of the U.S teens who said they have misrepresented their age online to gain access to websites and online services.

The $\widehat{p}=0.49$ means that, 49% out of 799 U.S. teens in the sample stated that they misrepresented their age online to gain access to websites and online services.  The proportion of the U.S teens who said they have misrepresented their age online to gain access to websites and online services is the best estimate of p.

b.

To determine

Find the standard error of $\widehat{p}$.

Answer to Problem 1E

The standard error of $\widehat{p}$ is 0.018.

Explanation of Solution

Calculation:

Standard error:

If the standard deviation of a sampling distribution is estimated by using statistics of the data, the estimate is termed as standard error.

$SE\left(\widehat{p}\right)=\sqrt{\frac{\widehat{p}\widehat{q}}{n}}$

Where, n be the number of samples, $\widehat{p}$ be the sample proportion and $\widehat{q}=1-\widehat{p}$.

Substitute $\widehat{p}$ as 0.49, $\widehat{q}$ as $0.51\left(=1-0.49\right)$ and n as 799 in the formula,

$\begin{array}{c}SE\left(\widehat{p}\right)=\sqrt{\frac{0.49\left(0.51\right)}{799}}\\ =\sqrt{\frac{0.2499}{799}}\\ =\sqrt{0.0003128}\\ \approx 0.018\end{array}$

Thus, the standard error of $\widehat{p}$ is 0.018.

c.

To determine

Explain what the standard error in this context.

Explanation of Solution

Interpretation:

Here the value of p is not known. Therefore $\widehat{p}$ is used to estimate the standard deviation of the sampling distribution.

When 799 teens are asked whether they misrepresented their age online to gain access to websites and online services, the standard error is the estimate that helps to know the amount of variation in the sample proportion between one sample to another sample.