Respuesta :
Answer:
(1) 97
(2) 385
(3) 9604
Step-by-step explanation:
The (1 - α) % confidence interval for population proportion is:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The margin of error in this interval is:
[tex]MOE= z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The formula to compute the sample size is:
[tex]\\n=\frac{z_{\alpha/2}^{2}\times \hat p(1-\hat p)}{MOE^{2}}[/tex]
(1)
Given:
[tex]\hat p = 0.50\\MOE=0.1\\z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]
*Use the z-table for the critical value.
Compute the value of n as follows:
[tex]\\n=\frac{z_{\alpha/2}^{2}\times \hat p(1-\hat p)}{MOE^{2}}\\=\frac{1.96^{2}\times0.50\times(1-0.50)}{0.1^{2}}\\=96.04\\\approx97[/tex]
Thus, the minimum sample size required is 97.
(2)
Given:
[tex]\hat p = 0.50\\MOE=0.05\\z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]
*Use the z-table for the critical value.
Compute the value of n as follows:
[tex]\\n=\frac{z_{\alpha/2}^{2}\times \hat p(1-\hat p)}{MOE^{2}}\\=\frac{1.96^{2}\times0.50\times(1-0.50)}{0.05^{2}}\\=384.16\\\approx385[/tex]
Thus, the minimum sample size required is 385.
(3)
Given:
[tex]\hat p = 0.50\\MOE=0.01\\z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]
*Use the z-table for the critical value.
Compute the value of n as follows:
[tex]\\n=\frac{z_{\alpha/2}^{2}\times \hat p(1-\hat p)}{MOE^{2}}\\=\frac{1.96^{2}\times0.50\times(1-0.50)}{0.01^{2}}\\=9604[/tex]
Thus, the minimum sample size required is 9604.
