Answer:
The 95% confidence interval for the difference in the population proportions( Pi - P2)
(0.2674 ,0.4055)
the upper bound for a 95% confidence interval for the difference in the population proportions, Pi - P2
0.4055
Step-by-step explanation:
Step :- (1)
Given data the results of the survey showed that among 620 non-smokers, 318 had said "yes"
The first proportion ![p_{1} = \frac{318}{620} =0.5129](https://tex.z-dn.net/?f=p_%7B1%7D%20%20%3D%20%5Cfrac%7B318%7D%7B620%7D%20%3D0.5129)
q₁ = 1- p₁ = 1-0.5129 =0.4871
Given data the results of the survey showed that among 195 smokers, 35 had said "yes".
The second proportion ![p_{2} = \frac{35}{195} =0.179](https://tex.z-dn.net/?f=p_%7B2%7D%20%20%3D%20%5Cfrac%7B35%7D%7B195%7D%20%3D0.179)
q₂ = 1- p₂ = 1-0.179 =0.821
<u>Step :-(2)</u>
<u>The 95% confidence interval for the difference in the population proportions( Pi - P2)</u>
(p₁-p₂ ± z₀.₉₅ se(p₁-p₂))
The standard error (p₁-p₂) is defined by
= ![\sqrt{\frac{p_{1}q_{1} }{n_{1} }+\frac{p_{2}q_{2} }{n_{2} } }](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7Bp_%7B1%7Dq_%7B1%7D%20%20%7D%7Bn_%7B1%7D%20%7D%2B%5Cfrac%7Bp_%7B2%7Dq_%7B2%7D%20%20%7D%7Bn_%7B2%7D%20%7D%20%20%7D)
= ![\sqrt{\frac{0.5129 X 0.4871 }{620 }+\frac{0.179 X 0.821 }{195 } }](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B0.5129%20X%200.4871%20%20%7D%7B620%20%7D%2B%5Cfrac%7B0.179%20X%200.821%20%7D%7B195%20%7D%20%20%7D)
= 0.0339
<u>The 95% confidence interval for the difference in the population proportions( Pi - P2)</u>
(p₁-p₂ ± z₀.₉₅ se(p₁-p₂))
(p₁-p₂ - z₀.₉₅ se(p₁-p₂) , p₁-p₂ + z₀.₉₅ se(p₁-p₂) )
(0.5129-0.179) - 1.96 × 0.0339 , 0.5129 -0.179) - 1.96 × 0.0339)
(0.339 -0.0665 , 0.339 +0.0665 )
(0.2674 ,0.4055)
<u>Conclusion</u>:-
The 95% confidence interval for the difference in the population proportions( Pi - P2)
(0.2674 ,0.4055)