Respuesta :
Answer:
(a) The sample size required is 2401.
(b) The sample size required is 2377.
(c) Yes, on increasing the proportion value the sample size decreased.
Step-by-step explanation:
The confidence interval for population proportion p is:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hatp(1-\hat p)}{n}}[/tex]
The margin of error in this interval is:
[tex]MOE=z_{\alpha/2}\sqrt{\frac{\hatp(1-\hat p)}{n}}[/tex]
The information provided is:
MOE = 0.02
[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]
(a)
Assume that the proportion value is 0.50.
Compute the value of n as follows:
[tex]MOE=z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\0.02=1.96\times \sqrt{\frac{0.50(1-0.50)}{n}}\\n=\frac{1.96^{2}\times0.50(1-0.50)}{0.02^{2}}\\=2401[/tex]
Thus, the sample size required is 2401.
(b)
Given that the proportion value is 0.55.
Compute the value of n as follows:
[tex]MOE=z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\0.02=1.96\times \sqrt{\frac{0.55(1-0.55)}{n}}\\n=\frac{1.96^{2}\times0.55(1-0.55)}{0.02^{2}}\\=2376.99\\\approx2377[/tex]
Thus, the sample size required is 2377.
(c)
On increasing the proportion value the sample size decreased.
