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].
- 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].
