Respuesta :
Answer:
The 99% confidence interval estimate for the difference between the percentage of women and men who prefer the beach over the mountains is -18.72% < (p₁ - p₂) < 4.72%
Step-by-step explanation:
The given parameters are;
Sample size of the women sample, n₁ = 200
Percentage of the women that prefer beach, p₁ = 45%, [tex]\hat p_1[/tex] = 0.45
Sample size of the men sample n₂ = 300
Percentage of the men that prefer beach, p₂ = 52%, [tex]\hat p_2[/tex] = 0.52
Confidence level of the confidence interval = 99%
α = 1 - 0.99 = 0.01, therefore, α/2 = 0.005
The equation for the confidence interval for the difference between two proportions is given as follows;
[tex]\left (\hat{p}_{1} - \hat{p}_{2} \right )\pm z_{\alpha /2}\sqrt{\dfrac{\hat{p}_{1}(1-\hat{p}_{1})}{n_{1}}+\dfrac{\hat{p}_{2}(1-\hat{p}_{2})}{n_{2}}}[/tex]
[tex]z_{\alpha /2}[/tex] = ±2.57 from the z score tables
Plugging in the vales, we have;
[tex]\left (0.45 - 0.52 \right )\pm 2.57\sqrt{\dfrac{0.45(1-0.45)}{200}+\dfrac{0.52(1-0.52)}{300}}[/tex]
Which gives;
-0.1872 < [tex](\hat p_1 - \hat p_2)[/tex] < 0.0472
The 99% confidence interval estimate for the difference between the percentage of women and men who prefer the beach over the mountains = -18.72% < (p₁ - p₂) < 4.72%.
