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Chapter 16: Problem 41

Death penalty, again In the survey on the death penalty you read about in the Step-by-Step Example, 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 \(53 \%\) 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?

Short Answer

Expert verified
a. The possible bias is the phrasing bias, which can lead respondents to choose the first option they hear. b. The \(95\%\) confidence interval for the proportion of the general public that believes the death penalty is being fairly applied is \(0.561, 0.599\). c. The margin of error is lower for the combined sample because as sample size increases, the margin of error decreases.

Step by step solution

01

Identify Bias

Here, the possible bias is the phrasing bias. This is due to the fact that the order of options in a question can impact the respondent's choice. In this case, by changing the order of 'fairly' and 'unfairly' in the question, it could lead respondents to choose the first option they hear, thus affecting the results of the survey.
02

Calculate Proportions

The first group had \(53\%\) of 510 respondents, which is 270.3 people, believing the death penalty is fairly applied. The second group had \(64\%\) of 510 respondents, which is 326.4 people, believing the same. The combined number of respondents for both groups is 596.7 out of 1020.
03

Calculate Confidence Interval

For a given confidence level of \(95\%\), the Z-score is approximately \(1.96\). The formula for the confidence interval is \[ \text{p} \pm \text{Z} \cdot \sqrt{\frac{\text{p}(1-\text{p})}{\text{n}}} \], where p is the sample proportion and n is the sample size. Substituting the values, we get \[ \frac{596.7}{1020} \pm 1.96 \cdot \sqrt{\frac{596.7/1020 (1-596.7/1020)}{1020}} \], which simplifies to \[ 0.58 \pm 0.019\]. So the \(95\%\) confidence interval is \(0.561, 0.599\).
04

Compare Margin of Errors

The margin of error is lower for the combined sample compared to separate groups. This is because as sample size increases, margin of error decreases. The larger data set provides a more accurate estimate of the population parameter, hence reducing the variability or the margin of error.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Phrasing Bias

Phrasing bias occurs when the wording or order of questions in a survey influences the responses of participants. This can happen when questions are leading, suggestive, or framed in a way that sways the respondent's answer. In the survey example regarding the death penalty, the order of 'fairly' and 'unfairly' was reversed for half of the respondents. This seemingly minor change can significantly impact the results, as some respondents may be influenced by the first option presented, which is known as a primacy effect. Ensuring neutral phrasing and a balanced order of options can help minimize this type of bias.

Understanding Sampling Bias

Sampling bias occurs when certain members of the intended population are systematically excluded from the sample, leading to a sample that's not truly representative. This can result from issues like non-random sampling methods, undercoverage, and non-response bias. Ensuring a truly random sample, where each individual in the population has an equal chance of being selected, is crucial to obtaining unbiased, generalizable results. Additionally, researchers need to address potential issues that could exclude segments of the population from the sample.

Examining Statistical Significance

Statistical significance is a measure of whether the results observed in a study or experiment are likely due to chance or if they reflect true differences or relationships. It's typically assessed with a p-value, which indicates the probability that the observed results would occur if there was no actual effect. A common threshold for significance is a p-value of 0.05 or lower, meaning there's a 5% probability (or less) that the results are due to random chance. In the context of confidence intervals, if the interval does not contain the null hypothesis value, the result is considered statistically significant.

Understanding Margin of Error

Margin of error is a statistic expressing the amount of random sampling error in a survey's results. It represents the range within which the true population parameter is expected to fall with a certain level of confidence. In simpler terms, it gives us the 'wiggle room' that exists around our sample estimate due to the fact that we are using a sample rather than a complete count of the population. As observed in the death penalty survey exercise, when we increase the sample size by combining groups, the margin of error decreases. This is because a larger sample size results in a more precise estimate of the population parameter, reducing uncertainty and therefore, the margin of error.

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Most popular questions from this chapter

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