91Ó°ÊÓ

Gender gap A presidential candidate fears he has a problem with women voters. His campaign staff plans to run a poll to assess the situation. They'Il randomly sample 300 men and 300 women, asking if they have a favorable impression of the candidate. Obviously, the staff can't know this, but suppose the candidate has a positive image with \(59 \%\) of males but with only \(53 \%\) of females. a) What sampling design is his staff planning to use? b) What difference would you expect the poll to show? c) Of course, sampling error means the poll won't reflect the difference perfectly. What's the standard deviation for the difference in the proportions? d) Sketch a sampling model for the size difference in proportions of men and women with favorable impressions of this candidate that might appear in a poll like this. e) Could the campaign be misled by the poll, concluding that there really is no gender gap? Explain.

Step by step solution

01

Identify Sampling Design

The staff is planning to use a stratified sampling design since they are separately sampling the population of men and women, which constitutes two distinct strata. Stratified sampling is useful when we want to ensure representation from each subgroup within a population.

02

Calculate Expected Difference

We are given that 59% of males have a favorable opinion and 53% of females do. The expected difference in proportions is, therefore: \[0.59 - 0.53 = 0.06\]So, we expect the poll to show a difference of 0.06 (6%) between men and women.

03

Calculate Standard Deviation for Difference

For the difference in proportions, the standard deviation can be computed using the formula: \[\text{SD}(p_1 - p_2) = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\]Substitute the given values: \[p_1 = 0.59, \ p_2 = 0.53, \ n_1 = n_2 = 300\]\[\text{SD}(0.59 - 0.53) = \sqrt{\frac{0.59 \times 0.41}{300} + \frac{0.53 \times 0.47}{300}} \approx 0.046\]The standard deviation is approximately 0.046.

04

Sketch Sampling Model

The sampling model for the difference in proportions of favorable impressions would be approximately normally distributed due to the large sample sizes (Central Limit Theorem). Centered around the expected difference of 0.06, with standard deviation 0.046, this model can be sketched as a normal distribution curve with mean 0.06 and spread defined by the standard deviation.

05

Evaluate Risk of Misleading Poll Result

Given a standard deviation of 0.046, if the poll indicated a difference of, say 0.02 or 0.03, this difference might not be statistically significant (falling within 1 standard deviation of the mean). Thus, there is a possibility the campaign staff could be misled into concluding there is no gender gap, particularly if the result is closer to zero than the expected 0.06.

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Key Concepts

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

Stratified Sampling

When trying to obtain accurate survey results, selecting the right sampling method is crucial. Stratified sampling is one technique used when a researcher wants to ensure that specific subgroups are adequately represented within a population. In the exercise, the campaign staff is using stratified sampling. They are separately sampling 300 men and 300 women, treating men and women as two distinct sub-groups or strata. This approach helps in capturing and analyzing differences among the subgroups, ensuring that both male and female perspectives are included in the poll results.

By using stratified sampling, the staff can more accurately deduce insights about public opinion by gender, as it minimizes the chances of underrepresented responses from either group. The benefit of this method is that it often provides more precise information than simply sampling the entire electorate without regard to gender.

Sampling Error

No matter how carefully a poll is conducted, there is always a degree of variability termed "sampling error." This is the difference between the actual population parameter and the sample statistic obtained from the poll.

This error arises because the sample is only a portion of the population and might not perfectly represent the entire group. In the exercise, even though the expected difference between favorable impressions from men and women is calculated at 6%, the actual result might vary due to sampling error.

Ensuring sample sizes are substantial, as the campaign staff have with 300 male and 300 female respondents, helps to minimize sampling error and provide results that are closer to the real values. This means while the exact percentage point difference might fluctuate slightly, the general findings should remain statistically sound.

Standard Deviation

Standard deviation is a measure of the dispersion or spread in a set of data values. It shows how much variation exists from the average (mean). When calculating the standard deviation for the difference in proportions, the formula accounts for the variability within each subgroup.

For this exercise, the standard deviation for the difference in proportions between men and women is approximately 0.046. This figure helps quantifying how the poll's observed differences might fluctuate around the expected difference of 0.06 (6%).

By understanding the standard deviation, campaign staff can determine whether any observed differences in polling results are significant or likely due to random chance. If the difference falls within one standard deviation of the mean, it may suggest that the gender gap observed in the poll is still consistent with the broader population's attitudes.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics. It states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.

In the scenario described, both sample sizes are large (300 men and 300 women), which ensures that the sampling distribution of their differences in favorable opinion will be approximately normal. This allows the creation of a sampling model represented by a normal curve, centered on the expected difference of 0.06 with a standard deviation of 0.046.

Thanks to the CLT, not only can we assume a normal distribution, but also use it to predict and interpret outcomes. It helps the campaign staff judge whether observed differences in poll results are meaningful or just products of chance within the context of the polling data.