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 the confidence interval for the difference in proportions, we can use the following formula:
Confidence Interval = (pA - pB) ± z * sqrt[(pA(1 - pA)/nA) + (pB(1 - pB)/nB)]
where
pA = proportion of people choosing "factories" in version A
pB = proportion of people choosing "factories" in version B
nA = number of respondents who used version A
nB = number of respondents who used version B
z = z-score associated with the desired confidence level
(a) Let's first substitute the given values into the formula:
pA = 170/460 = 0.3696 (approximately)
pB = 141/440 = 0.3205 (approximately)
nA = 460
nB = 440
z = 1.96 (for 95% confidence level)
Now, plug the values into the formula:
Confidence Interval = (0.3696 - 0.3205) ± 1.96 * sqrt[((0.3696 * (1 - 0.3696))/460) + ((0.3205 * (1 - 0.3205))/440)]
Calculating the confidence interval:
Confidence Interval = 0.0491 ± 1.96 * sqrt[(0.3680/460) + (0.2152/440)]
Confidence Interval = 0.0491 ± 1.96 * sqrt(0.0008 + 0.0005)
Confidence Interval = 0.0491 ± 1.96 * sqrt(0.0013)
Confidence Interval = 0.0491 ± 1.96 * 0.0361
Confidence Interval = 0.0491 ± 0.0707
Therefore, the 95% confidence interval for pA - pB is (0.0491 - 0.0707) to (0.0491 + 0.0707), which can be simplified to (-0.0216, 0.1198).
(b) To determine if the versions are consistent concerning this answer, we need to check if the confidence interval includes zero. If zero is within the confidence interval, it means the proportions are not significantly different. If zero is not within the confidence interval, it means the proportions are significantly different.
In this case, when we look at the 95% confidence interval (-0.0216, 0.1198), we note that zero is within the interval. Therefore, we can conclude that the versions are consistent concerning the major cause of air pollution, as the proportions of people choosing "factories" are not significantly different between version A and B.