Respuesta :
Answer:
(a) The best point estimate of the population proportion p is 0.51.
(b) The margin of error is 0.0299.
(c) The 95% confidence interval for population proportion of all respondent who said "yes" is (0.4801, 0.5399).
Step-by-step explanation:
The random variable X is defined as the number of respondents who felt vulnerable to identity theft.
The information provided is:
n = 1074
x = 543
(a)
A point estimate of a parameter is a distinct value used for the estimation the parameter. For instance, the sample mean [tex]\bar x[/tex] is a point estimate of the population mean μ.
The best point estimate of the population proportion p is the sample proportion [tex]\hat p[/tex].
Compute the sample proportion value as follows:
[tex]\hat p=\frac{x}{n}=\frac{543}{1074}=0.505586\approx0.51[/tex]
Thus, the best point estimate of the population proportion p is 0.51.
(b)
The margin of error for the confidence interval of p is:
[tex]E=z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
For 95% confidence level the critical value of z is:
[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]
Compute the margin of error as follows:
[tex]E=z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}=1.96\sqrt{\frac{0.51(1-0.51)}{1074}}=0.0299[/tex]
Thus, the margin of error is 0.0299.
(c)
Construct the 95% confidence interval for population proportion as follows:
[tex]CI=\hat p\pm E\\=0.51\pm0.0299\\=(0.51-0.0299, 0.51+0.0299)\\=(0.4801, 0.5399)[/tex]
Thus, the 95% confidence interval for population proportion of all respondent who said "yes" is (0.4801, 0.5399).
