91Ó°ÊÓ

Candidates for political office realize that different levels of support among men and women may be a crucial factor in determining the outcome of an election. One candidate finds that \(52 \%\) of 473 men polled say they will vote for him, but only \(45 \%\) of the 522 women in the poll express support. a. Write a \(95 \%\) confidence interval for the percent of male voters who may vote for this candidate. Interpret your interval. b. Write a \(95 \%\) confidence interval for the percent of female voters who may vote for him. Interpret your interval. c. Do the intervals for males and females overlap? What do you think this means about the gender gap? d. Find a \(95 \%\) confidence interval for the difference in the proportions of males and females who will vote for this candidate. Interpret your interval. e. Does this interval contain zero? What does that mean? f. Why do the results in parts \(c\) and e seem contradictory? If we want to see if there is a gender gap among voters with respect to this candidate, which is the correct approach? Why?

Step by step solution

01

Confidence Interval for Male Voters

To calculate the \(95 \%\) confidence interval for the proportion of male voters who may vote for this candidate, we use the formula for a confidence interval: \[ p \pm Z * \sqrt{p*(1-p)/n} \], where p is the sample proportion, \(Z\) is the z-score associated with our desired level of confidence (1.96 for \(95 \%\) confidence), and n is the sample size. Here, \( p = 52/100 \) and \( n = 473 \). Plugging these numbers into the formula will give us a confidence interval.

02

Confidence Interval for Female Voters

We use the same formula as described in step 1, this time applying it to the female voter data. Here we have \( p = 45/100 \) and \( n = 522 \). Calculate this to get a confidence interval.

03

Comparison of Intervals

Examine the two confidence intervals just calculated. If there is any overlap between the two, it suggests that the difference between the two observed proportions may not be significant.

04

Confidence Interval for the Difference

To find the \(95 \%\) confidence interval for the difference, we use a similar formula: \[ (p1 - p2) \pm Z * \sqrt{p1*(1-p1)/n1 + p2*(1-p2)/n2 } \], where \(p1\) and \(p2\) are the proportions of male and female voters respectively, \(n1\) and \(n2\) are the sizes of the male and female samples respectively, and \(Z\) is the z-score.

05

Interpretation

If the confidence interval calculated in Step 4 contains zero, it signifies that there is a possibility that there is no real difference in voting patterns between males and females.

06

Addressing the Contradiction

Explain why the results from Steps 3 and 5 appear contradictory and which approach is better in determining a gender gap.

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Key Concepts

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

Statistical Significance

Statistical significance refers to the likelihood that the result of an analysis or experiment is caused by something other than chance. This concept often determines whether or not researchers reject a null hypothesis in studies, indicating that observed results are not random but reflect true effects or relationships.

When looking at the gender gap in voting, a statistically significant result would mean that the difference in voting patterns between men and women for a particular candidate is unlikely to have occurred by random chance alone. Researchers use p-values and confidence intervals to assess this significance. A commonly accepted threshold for significance is a p-value of less than 0.05 or a 95% confidence interval that does not include 0 when assessing difference between groups.

Proportion

The proportion, in statistics, is a type of ratio that represents a part of a whole. It is typically expressed as a percentage and is calculated by dividing the number of outcomes of interest by the total number of possible outcomes. In the context of voting behavior, proportion might refer to the percentage of voters in a sample who support a particular candidate.

Calculating the proportion of male and female voters who support a candidate, as in the exercise example, gives us insight into the candidate's popularity within these groups. Understanding and calculating proportions is fundamental to analyzing demographic data and interpreting polling results.

Z-score

The z-score is a statistical measure that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. In the context of confidence intervals, the z-score is used to determine how far away the boundary of the interval is from the sample proportion.

For a 95% confidence interval, the z-score is approximately 1.96, indicating that the interval range extends 1.96 standard deviations from the sample mean on either side. This z-score helps capture the range within which the true population proportion is likely to fall with 95% confidence.

Sample Size

Sample size is crucial in statistical analysis because it influences the precision of the estimated proportions or means. A larger sample size typically reduces the margin of error and yields more reliable results, as it should better represent the population. In the context of the exercise, the sample sizes for the male and female populations are substantial enough to provide reasonably precise estimates of voter support.

However, it's important to recognize that increasing sample size improves confidence interval precision only up to a point. Researchers must balance the benefits of a larger sample with the resources required to collect the data.

Gender Gap in Voting

The gender gap in voting refers to differences in voting behavior between men and women. This can manifest in varying levels of support for candidates, parties, or issues. In polling data, as highlighted by the problem statement, the gender gap is often investigated through comparisons of proportions — i.e., what percentage of men versus women support a particular candidate.

Analyzing the gender gap is complex and involves looking at various factors, including the statistical significance of the observed differences, the proportions within each gender group, and the sizes of the samples. The correct interpretation of the gender gap involves comparing confidence intervals for each group and for the difference between groups to see if a substantive gap exists.