Question

Construct a confidence interval for P1 - P2 at the given level of confidence. Xy = 33, ny = 265, X2 = 30, n2 = 283, 90% confidence and : The researchers are % confident the difference between the two population proportions, P1-P2, is between (Use ascending order. Type an integer or decimal rounded to three decimal places as needed.)

Answer 1

Solution :

Given that,

$\hat p$₁ = x₁ / n₁ = 0.125

$\hat p$₂ = x₂ / n₂ = 0.106

1) Point estimate of difference between two proportions

= $\hat p$ ₁ -  $\hat p$₂

= 0.125 - 0.106

= 0.019

2)

Our aim is to construct 90% confidence interval.

$\therefore$ c = 0.90

$\therefore$$\alpha$ = 1- c = 1- 0.90 = 0.10

$\therefore$  $\alpha$/2 = 0.10 $\slash$ 2 = 0.05 and 1- $\alpha$ /2 = 0.950

$\therefore$$Z_{\alpha/2}$ = 1.645 (use z table)

Margin of error =  $Z_{\alpha/2}$ * $\sqrt{\frac{\hat p_{1}(1-\hat p_{1})}{n_{1}}+\frac{\hat p_{2}(1-\hat p_{2})}{n_{2}}}$

= $1.645 * \sqrt{\frac{0.125(1-0.125)}{265}+\frac{0.106(1-0.106)}{283}}$

= 0.0445

3) Required interval is

Point estimate  $\pm$ Margin of error

0.019 $\pm$   0.0445

( 0.019 - 0.0445 , 0.019 +  0.0445 )

(-0.026 , 0.063)