Pew Research in 2018 polled a random sample of 1058 U.S. teens (ages 13-17) about Internet use. reported that they were online “almost constantly” -a fact of great interest to advertisers. Complete parts a through c below.
a) Explain the meaning of in the context of this situation. Choose the correct answer below.
A. The meaning of is that of the 1058 teens in the sample reported that they were online “almost constantly.” This is the researchers’ best estimate of , the proportion of all U.S. teens who would say that.
B. The meaning of is that of the 1058 teens in the sample reported that they were online “almost constantly.” This is the researchers’ best estimate of , the proportion of the 1058 teens in the sample who said that.
C. The meaning of is that the researchers have confidence that of all U.S. teens would report that they were online “almost constantly.”
D. The meaning of is that the researchers have confidence that of the 1058 teens in the sample reported that they were online “almost constantly.”
b) Calculate the standard error of .

(Round to five decimal places as needed.)

Ask by Sherman Mathis. in the United States

Mar 31,2025

Question

Pew Research in 2018 polled a random sample of 1058 U.S. teens (ages 13-17) about Internet use. reported that they were online “almost constantly” -a fact of great interest to advertisers. Complete parts a through c below.
a) Explain the meaning of in the context of this situation. Choose the correct answer below.
A. The meaning of is that of the 1058 teens in the sample reported that they were online “almost constantly.” This is the researchers’ best estimate of , the proportion of all U.S. teens who would say that.
B. The meaning of is that of the 1058 teens in the sample reported that they were online “almost constantly.” This is the researchers’ best estimate of , the proportion of the 1058 teens in the sample who said that.
C. The meaning of is that the researchers have confidence that of all U.S. teens would report that they were online “almost constantly.”
D. The meaning of is that the researchers have confidence that of the 1058 teens in the sample reported that they were online “almost constantly.”
b) Calculate the standard error of .

(Round to five decimal places as needed.)

Ask by Sherman Mathis. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

**a)**  
$$ \hat{p}=0.447 $$ means that $$ 44.7\% $$ of the 1058 teens in the sample reported being online "almost constantly." This sample proportion is used as the researchers’ best estimate of $$ p $$, the true proportion of all U.S. teens who would report being online "almost constantly."  
The correct answer is **A**.

**b)**  
The standard error of $$ \hat{p} $$ is given by the formula:  
$$
\operatorname{SE}(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
$$
Substitute $$ \hat{p} = 0.447 $$ and $$ n = 1058 $$:
$$
\operatorname{SE}(\hat{p}) = \sqrt{\frac{0.447 \times (1-0.447)}{1058}} = \sqrt{\frac{0.447 \times 0.553}{1058}}
$$
First, calculate the numerator:
$$
0.447 \times 0.553 \approx 0.24719
$$
Then divide by $$ n $$:
$$
\frac{0.24719}{1058} \approx 0.00023367
$$
Now, take the square root:
$$
\operatorname{SE}(\hat{p}) \approx \sqrt{0.00023367} \approx 0.01530
$$
Thus, the standard error is approximately $$ 0.01530 $$.
**a)** $$ \hat{p}=0.447 $$ means that $$ 44.7\% $$ of the 1058 teens in the sample said they were online "almost constantly." This is the researchers' best estimate of the true proportion of all U.S. teens who would say the same. **b)** The standard error of $$ \hat{p} $$ is approximately $$ 0.01530 $$.

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