Question

Construct a confidence interval for p1−p2 at the given level of confidence.

x1=365

n1=536

x2=435

n2=593

90​% confidence

The researchers are (blank) % confident the difference between the two population​ proportions,

p1−p2​, is between (blank) and (blank)

Answer 1

$p_{1}=\frac{x_{1}}{n_{1}}=\frac{365}{536}=0.6810,p_{2}=\frac{x_{2}}{n_{2}}=\frac{435}{593}=0.7336$

We know that the 90% confidence interval for difference of proportions is given by:

$[(p_{1}-p_{2})\pm Z_{0.10/2}\sqrt{\frac{p_{1}(1-p_{1})}{n_{1}}+\frac{p_{2}(1-p_{2})}{n_{2}}}]$

$[(0.6810-0.7336)\pm Z_{0.05}\sqrt{\frac{0.6810(1-0.6810)}{536}+\frac{0.7336(1-0.7336)}{593}}]$

$[(0.6810-0.7336)\pm (1.645)\sqrt{\frac{0.6810(1-0.6810)}{536}+\frac{0.7336(1-0.7336)}{593}}]$

$[-0.0526\pm 0.0446]$

$[-0.0972,-0.0080]$

So, the researchers are 90% confident the difference between the two population​ proportions,

p1−p2​, is between -0.0972 and -0.0080.