An environmental survey contained a question asking what respondents
thought was the major cause of air pollution in this country, giving the choices “automobiles,” “factories,” and “incinerators.” Two versions of the test, A and B, were used. Let p A and p B be the respective proportions of people using forms A and B who select “factories.” If 170 out of 460 people who used version A chose “factories” and
141 out of 440 people who used version B chose “factories,”
(a) Find a 95% confidence interval for pA – p B .
(b) Do the versions seem to be consistent concerning this answer? Why or why not?
To find a 95% confidence interval for the difference between pA and pB, we can use the formula:
CI = (pA - pB) ± Z * √(p̂A(1 - p̂A) / nA + p̂B(1 - p̂B) / nB)
where:
- p̂A is the proportion of people in group A who chose "factories" (170/460)
- p̂B is the proportion of people in group B who chose "factories" (141/440)
- nA is the sample size for group A (460)
- nB is the sample size for group B (440)
- Z is the critical z-score for a 95% confidence interval (approximately 1.96)
Let's calculate the confidence interval step by step:
Step 1: Calculate p̂A and p̂B
p̂A = 170/460 ≈ 0.3696
p̂B = 141/440 ≈ 0.3205
Step 2: Calculate the standard error
SE = √(p̂A(1 - p̂A) / nA + p̂B(1 - p̂B) / nB)
SE = √((0.3696*(1 - 0.3696))/460 + (0.3205*(1 - 0.3205))/440)
SE ≈ 0.0365
Step 3: Calculate the margin of error
ME = Z * SE
ME = 1.96 * 0.0365
ME ≈ 0.0715
Step 4: Calculate the lower and upper bounds of the confidence interval
Lower bound = (pA - pB) - ME
Upper bound = (pA - pB) + ME
Lower bound = (0.3696 - 0.3205) - 0.0715 ≈ 0.021------------------------------------------------ (1)
Upper bound = (0.3696 - 0.3205) + 0.0715 ≈ 0.121------------------------------------------------ (2)
So, the 95% confidence interval for pA - pB is approximately 0.021 to 0.121.
(b) To determine if the versions are consistent concerning this answer, we can check if the confidence interval includes zero. If the confidence interval includes zero, it suggests that there is no significant difference between the proportions.
Looking at the calculated interval (0.021 to 0.121), we can see that it does not include zero. Therefore, we can conclude that the versions A and B are not consistent concerning the answer to the question. There is a significant difference in the proportions of people choosing "factories" between the two versions.