Question

Time magazine reported the result of a telephone poll of randomly selected adult Americans. The poll asked these adult Americans: "Should the federal tax on cigarettes be raised to pay for health care reform?" The results of the survey showed that among 620 non-smokers, 318 had said "yes" and among 195 smokers, 35 had said "yes".

We want to estimate the actual difference between the proportions of non-smokers and smokers who said "yes" to federal tax increase. Notation: 1=non-smokers and 2-smokers. Based on this data, what is the upper bound for a 95% confidence interval for the difference in the population proportions, Pi - P2?

Answer 1

The 95% confidence interval for difference in proportion of non-smokers and smokers who said "yes" to federal tax increase is (0.26, 0.40).

Explanation:

The confidence interval for difference in proportion formula is,

`CI=(\hat p_(1)-\hat p_(2))\pm z_(\alpha/2)* \sqrt{(\hat p_(1)(1-\hat p_(1)))/(n_(1))+(\hat p_(2)(1-\hat p_(2)))/(n_(2))}`

The given information is:

`n_(1)=620\\n_(2)=195\\X_(1)=318\\X_(2)=35`

Compute the sample proportions as follows:

`\hat p_(1)=(X_(1))/(n_(1))=(318)/(620)=0.51\\\\\hat p_(2)=(X_(2))/(n_(2))=(35)/(195)=0.18`

Compute the critical value of z for the 95% confidence level as follows:

`z_(\alpha/2)=z_(0.05/2)=z_(0.025)=1.96`

*Use the standard normal table.

Compute the 95% confidence interval for difference between proportions as follows:

`CI=(\hat p_(1)-\hat p_(2))\pm z_(\alpha/2)* \sqrt{(\hat p_(1)(1-\hat p_(1)))/(n_(1))+(\hat p_(2)(1-\hat p_(2)))/(n_(2))}`

`=(0.51-0.18)\pm 1.96* \sqrt{(0.51(1-0.51))/(620)+(0.18(1-0.18))/(195)}`

`=0.33\pm 0.067\\=(0.263, 0.397)\\\approx (0.26, 0.40)`

Thus, the 95% confidence interval for difference in proportion of non-smokers and smokers who said "yes" to federal tax increase is (0.26, 0.40).

Answer 2

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

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) = (318)/(620) =0.5129`

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) = (35)/(195) =0.179`

q₂ = 1- p₂ = 1-0.179 =0.821

Step :-(2)

The 95% confidence interval for the difference in the population proportions( Pi - P2)

(p₁-p₂ ± z₀.₉₅ se(p₁-p₂))

The standard error (p₁-p₂) is defined by

=
`\sqrt{(p_(1)q_(1) )/(n_(1) )+(p_(2)q_(2) )/(n_(2) ) }`

=
`\sqrt{(0.5129 X 0.4871 )/(620 )+(0.179 X 0.821 )/(195 ) }`

= 0.0339

The 95% confidence interval for the difference in the population proportions( Pi - P2)

(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)

Conclusion:-

The 95% confidence interval for the difference in the population proportions( Pi - P2)

(0.2674 ,0.4055)