In the chapter example and again in given Exercise we looked at a Gallup Poll investigating the public s attitude toward the death penalty. In response to one question, 60% thought it was fair, but when the question was phrased differently, the proportion in favor dropped to 54%.
a) What kind of bias may be present here?
b) Each group consisted of 510 respondents. If we combine them, considering the overall group to be one larger random sample, what is a 95% confidence interval for the proportion of the general public that thinks the death penalty is being fairly applied?
c) How does the margin of error based on this pooled sample compare with the margins of error from the separate groups? Why?
Exercise :
Death penalty, again. In the survey on the death penalty you read about in the chapter, the Gallup Poll actually split the sample at random, asking 510 respondents the question quoted earlier, Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today? The other 510 were asked Generally speaking, do you believe the death penalty is applied unfairly or fairly in this country today? Seems like the same question, but sometimes the order of the choices matters. Suppose that for the second way of phrasing it, only 54% said they thought the death penalty was fairly applied.
a) Construct a 95% confidence interval for the true proportion of adults who approve of the way the death penalty is currently applied, according to the responses for this second question.
b) Recall that 60% of the respondents in the other random half of the study said that the death penalty is applied fairly. Does a proportion of 0.60 fall inside the confidence interval you just found?