Answer:
(a) The sample sizes are 6787.
(b) The sample sizes are 6666.
Step-by-step explanation:
(a)
The information provided is:
Confidence level = 98%
MOE = 0.02
n₁ = n₂ = n
[tex]\hat p_{1} = \hat p_{2} = \hat p = 0.50\ (\text{Assume})[/tex]
Compute the sample sizes as follows:
[tex]MOE=z_{\alpha/2}\times\sqrt{\frac{2\times\hat p(1-\hat p)}{n}[/tex]
[tex]n=\frac{2\times\hat p(1-\hat p)\times (z_{\alpha/2})^{2}}{MOE^{2}}[/tex]
[tex]=\frac{2\times0.50(1-0.50)\times (2.33)^{2}}{0.02^{2}}\\\\=6786.125\\\\\approx 6787[/tex]
Thus, the sample sizes are 6787.
(b)
Now it is provided that:
[tex]\hat p_{1}=0.45\\\hat p_{2}=0.58[/tex]
Compute the sample size as follows:
[tex]MOE=z_{\alpha/2}\times\sqrt{\frac{\hat p_{1}(1-\hat p_{1})+\hat p_{2}(1-\hat p_{2})}{n}[/tex]
[tex]n=\frac{(z_{\alpha/2})^{2}\times [\hat p_{1}(1-\hat p_{1})+\hat p_{2}(1-\hat p_{2})]}{MOE^{2}}[/tex]
[tex]=\frac{2.33^{2}\times [0.45(1-0.45)+0.58(1-0.58)]}{0.02^{2}}\\\\=6665.331975\\\\\approx 6666[/tex]
Thus, the sample sizes are 6666.
