Answer: [tex]\mu_{\hat{p}}=0.14[/tex] and [tex]\sigma_{\hat{p}}=0.0347[/tex]
Step-by-step explanation:
In sampling distribution of [tex]\hat{p}[/tex].
The mean and standard deviation of the sampling distribution of p is given by :-
[tex]\mu_{\hat{p}}=p[/tex]
[tex]\sigma_{\hat{p}}=\sqrt{\dfrac{p(1-p)}{n}}[/tex] , where p= population proportion and n= sample size.
Let p be the population proportion of unmarried couples in the United States are mixed racially or ethnically.
As per given , we have
n = 100
p= 14% =0.14
Then, the mean and standard deviation of the sampling distribution of p will be :
[tex]\mu_{\hat{p}}=0.14[/tex]
[tex]\sigma_{\hat{p}}=\sqrt{\dfrac{0.14(1-0.14)}{100}}[/tex]
[tex]\sigma_{\hat{p}}=\sqrt{\dfrac{0.14(0.86)}{100}}[/tex]
[tex]\sigma_{\hat{p}}=\sqrt{\dfrac{0.1204}{100}}[/tex]
[tex]\sigma_{\hat{p}}=\sqrt{0.001204}[/tex]
[tex]\sigma_{\hat{p}}=0.0346987031458\approx0.0347[/tex]
Hence, the required answer : [tex]\mu_{\hat{p}}=0.14[/tex] and [tex]\sigma_{\hat{p}}=0.0347[/tex]
