Smartphones: A poll agency reports that [tex]$80 \%$[/tex] of teenagers aged [tex]$12-17$[/tex] own smartphones. A random sample of 250 teenagers is drawn. Round your answers to four decimal places as needed.

Part: [tex]$0 / 6$[/tex] [tex]$\square$[/tex]

Part 1 of 6
(a) Find the mean [tex]$\mu_{\hat{p}}$[/tex].

The mean [tex]$\mu_{\hat{p}}$[/tex] is 0.8000. [tex]$\square$[/tex]

Part: [tex]$1 / 6$[/tex] [tex]$\square$[/tex]

Part 2 of 6
(b) Find the standard deviation [tex]$\sigma_{\hat{p}}$[/tex].

The standard deviation [tex]$\sigma_{\hat{p}}$[/tex] is [tex]$\square$[/tex]. [tex]$\square$[/tex]

[tex]$\square$[/tex]



Answer :

To solve this problem, let's start by understanding the parameters given:

- The proportion of teenagers owning smartphones ([tex]\( p \)[/tex]) is [tex]\( 0.80 \)[/tex] or 80%.
- The sample size ([tex]\( n \)[/tex]) is [tex]\( 250 \)[/tex].

Given these values, we can follow the steps to find the mean and standard deviation of the sampling distribution of the sample proportion ( [tex]\( \hat{p} \)[/tex]).

### Part 1: Finding the Mean [tex]\( \mu_{\hat{p}} \)[/tex]

The mean [tex]\( \mu_{\hat{p}} \)[/tex] of the sampling distribution of the sample proportion is equal to the population proportion [tex]\( p \)[/tex].

Therefore,
[tex]\[ \mu_{\hat{p}} = p = 0.80 \][/tex]

So, the mean [tex]\( \mu_{\hat{p}} \)[/tex] rounded to four decimal places is:
[tex]\[ \mu_{\hat{p}} = 0.8000 \][/tex]

### Part 2: Finding the Standard Deviation [tex]\( \sigma_{\hat{p}} \)[/tex]

The standard deviation [tex]\( \sigma_{\hat{p}} \)[/tex] of the sampling distribution of the sample proportion can be calculated using the formula:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{p (1 - p)}{n}} \][/tex]

Substitute the given values into the formula:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{0.80 \times (1 - 0.80)}{250}} \][/tex]

Simplify within the square root:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{0.80 \times 0.20}{250}} \][/tex]
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{0.16}{250}} \][/tex]
[tex]\[ \sigma_{\hat{p}} = \sqrt{0.00064} \][/tex]
[tex]\[ \sigma_{\hat{p}} = 0.0253 \][/tex]

So, the standard deviation [tex]\( \sigma_{\hat{p}} \)[/tex] rounded to four decimal places is:
[tex]\[ \sigma_{\hat{p}} = 0.0253 \][/tex]

### Final Answers

- The mean [tex]\( \mu_{\hat{p}} \)[/tex] is [tex]\( 0.8000 \)[/tex].
- The standard deviation [tex]\( \sigma_{\hat{p}} \)[/tex] is [tex]\( 0.0253 \)[/tex].