Respuesta :
Answer:
(a) The point estimate for the population proportion p is 0.34.
(b) The margin of error for the 99% confidence interval of population proportion p is 0.055.
(c) The 99% confidence interval of population proportion p is (0.285, 0.395).
Step-by-step explanation:
A point estimate of a parameter (population) is a distinct value used for the estimation the parameter (population). For instance, the sample mean [tex]\bar x[/tex] is a point estimate of the population mean μ.
Similarly, the the point estimate of the population proportion of a characteristic, p is the sample proportion [tex]\hat p[/tex].
The (1 - α)% confidence interval for the population proportion p is:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The margin of error for this interval is:
[tex]MOE= z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The information provided is:
[tex]\hat p=0.34\\n=500\\(1-\alpha)\%=99\%[/tex]
(a)
Compute the point estimate for the population proportion p as follows:
Point estimate of p = [tex]\hat p[/tex] = 0.34
Thus, the point estimate for the population proportion p is 0.34.
(b)
The critical value of z for 99% confidence level is:
[tex]z={\alpha/2}=z_{0.01/2}=z_{0.005}=2.58[/tex]
*Use a z-table for the value.
Compute the margin of error for the 99% confidence interval of population proportion p as follows:
[tex]MOE= z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
[tex]=2.58\sqrt{\frac{0.34(1-0.34)}{500}}[/tex]
[tex]=2.58\times 0.0212\\=0.055[/tex]
Thus, the margin of error for the 99% confidence interval of population proportion p is 0.055.
(c)
Compute the 99% confidence interval of population proportion p as follows:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
[tex]CI=\hat p\pm MOE[/tex]
[tex]=0.34\pm 0.055\\=(0.285, 0.395)[/tex]
Thus, the 99% confidence interval of population proportion p is (0.285, 0.395).
