Complete Question
The complete question is shown on the first uploaded image
Answer:
The interval is [tex]-0.0037 < p_1-p_2<-0.0023[/tex]
Step-by-step explanation:
From the question we are told that
The first sample size is [tex]n _1 = 1068000[/tex]
The first sample proportion is [tex]\r p_1 = 0.089[/tex]
The second sample size is [tex]n_2 = 1476000[/tex]
The second sample proportion is [tex]\r p_2 = 0.092[/tex]
Given that the confidence level is 95% then the level of significance is mathematically evaluated as
[tex]\alpha = (100 - 95 )\%[/tex]
[tex]\alpha = 0.05[/tex]
Next we obtain the critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table
The value is
[tex]Z_{\frac{\alpha }{2} } =z_c= 1.96[/tex]
Generally the 95% confidence interval is mathematically represented as
[tex](\r p_1 - \r p_2 ) -z_c \sqrt{ \frac{\r p_1 \r q_1 }{n_1} + \frac{\r p_2 \r q_2 }{n_2}} < (p_1 - p_2 ) < (\r p_1 - \r p_2 ) +z_c \sqrt{ \frac{\r p_1 \r q_1 }{n_1} + \frac{\r p_2 \r q_2 }{n_2}}[/tex]
Here [tex]\r q_1[/tex] is mathematically evaluated as [tex]\r q_1 = (1 - \r p_1)= 1-0.089 =0.911[/tex]
and [tex]\r q_2[/tex] is mathematically evaluated as [tex]\r q_2 = (1 - \r p_2) = 1- 0.092 = 0.908[/tex]
So
[tex](0.089 - 0.092 ) -1.96 \sqrt{ \frac{0.089* 0.911 }{1068000} + \frac{0.092* 0.908 }{1476000}} < (p_1 - p_2 ) < (0.089 - 0.092 ) +1.96 \sqrt{ \frac{0.089* 0.911 }{1068000} + \frac{0.092* 0.908 }{1476000}}[/tex]
[tex]-0.0037 < p_1-p_2<-0.0023[/tex]
