In the survey on the death penalty you read about in the...

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, $64 \%$ said they thought the death penalty was fairly applied. (Recall that $58 \%$ of the original 510 thought the same thing.) a) What kind of bias may be present here? b) If we combine them, considering the overall group to be one larger random sample of 1020 respondents, 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?

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, $64 \%$ said they thought the death penalty was fairly applied. (Recall that $58 \%$ of the original 510 thought the same thing.)
a) What kind of bias may be present here?
b) If we combine them, considering the overall group to be one larger random sample of 1020 respondents, 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?

Key Concepts

Response Order Bias

This concept refers to the idea that the order in which response options are presented in a survey can influence respondents’ answers. Even when the wording is logically equivalent, respondents may be more likely to choose the first option mentioned due to factors such as primacy effects. This bias can skew survey results and is an important consideration in survey design and interpretation.

Confidence Interval for a Proportion

A confidence interval for a proportion provides a range of values within which the true population proportion is expected to lie, with a specified level of confidence (commonly 95%). In the context of survey data, it’s calculated by taking the point estimate (the proportion from the sample) and adding and subtracting a margin of error, which is typically derived from the standard error of the proportion. This interval quantifies the uncertainty of the sample estimate in relation to the true population value.

Pooled Sample Analysis and Margin of Error

When data from two independent samples are combined, the analysis uses a larger overall sample size which typically leads to a smaller margin of error. The margin of error decreases because the standard error of the overall estimate is inversely related to the square root of the sample size. This pooled analysis provides a more precise estimate of the population proportion, reflecting less uncertainty than estimates derived from smaller, separate samples.

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