Gender gap Candidates for political office realize that...

Gender gap 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 and interpret a 95% confidence interval for the percent of female voters who may vote for him. 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?

Gender gap 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 and interpret a 95% confidence interval for the percent of female voters who may vote for him.
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?

Key Concepts

Appropriate Comparison Methodology

The correct approach to evaluating a gender gap is to compute and interpret the confidence interval for the difference between the proportions rather than simply checking whether the individual confidence intervals for each group overlap. This direct comparison accurately assesses if the observed difference in support is significant, avoiding the pitfalls associated with comparing separate intervals.

Statistical Significance and Zero in the Interval

When constructing a confidence interval for the difference between two proportions, if zero is not included in the interval, it indicates that there is a statistically significant difference between the groups at the given confidence level. This concept connects the idea of confidence intervals to hypothesis testing, where a zero difference would imply no effect or no gender gap.

Overlap of Confidence Intervals

The examination of overlapping confidence intervals may provide an intuitive sense of whether two estimates might be different, but overlapping intervals can sometimes be misleading. This is because individual confidence intervals for each proportion do not directly address the comparison between groups. A more precise evaluation involves directly calculating a confidence interval for the difference in proportions.

Difference in Proportions

This concept refers to comparing two independent proportions by constructing a confidence interval for their difference. The approach involves calculating the difference between the sample proportions and determining the standard error for this difference. The resulting confidence interval gauges whether there is a statistically significant difference between the two groups’ proportions by checking if zero is included.

Interpretation of Confidence Intervals

Interpreting a confidence interval means understanding that if the same sampling process were repeated many times, a certain percentage (e.g., 95%) of the calculated intervals would contain the true population parameter. For proportion estimates, this provides information about both the estimate and its potential error, allowing us to make probabilistic statements about the candidate’s support within a specified population.

Confidence Interval for a Proportion

This concept involves estimating an interval around a sample proportion to capture the true population proportion with a certain level of confidence (e.g., 95%). It uses the sample proportion and its standard error to compute a margin of error, and the confidence interval is formed by adding and subtracting this margin from the observed proportion. This method helps account for sampling variability and gives a range in which the true proportion is likely to lie.

Transcript

00:01 So we're dealing with confidence intervals for proportions in this question.

00:06 And so for our first one, we know that the number of males that will vote for a candidate is 52%.

00:14 And our formula for finding a confidence interval for proportions, a single proportion, is to take our p hat, which is our sample proportion, times the z star value, times the square root of the p hat, one minus the p hat.

00:31 Over end.

00:32 So let's do that for this mail.

00:34 So we know that the males, we have 52 % is what they estimate the proportion to be.

00:39 And we have our 95%.

00:42 So we're going to use that 1 .96.

00:44 And we have 0 .52 times 0 .48.

00:49 And then our sample size is given to us as 473.

00:54 So let's type that into our calculator.

00:57 And i have the 0 .52.

01:00 And i have the 0 .52.

01:01 I'll do the minus first to get the low end, 1 .96 times the square root of and 0 .52 times 0 .48 divided by that 473, and that's all underneath the radical, and i'll hit enter.

01:19 And i get the low interval, which i'll convert to a percent, is 47.

01:25 I'd come out to be 50 percent.

01:27 And the high, go back and change that subtraction sign into an addition sign.

01:35 Just got to back my calculator up.

01:37 And boom, got it.

01:39 And 56 .5%.

01:42 So basically, we're 95 % confident that the actual proportion is somewhere in here.

01:50 We recognize that when we sample that this value that we get as a point estimate is probably not the accurate value.

01:57 So when we do these calculations and doing a 95 % confidence interval, 95 % of calculated intervals will actually have the true proportion for males, which we'd call, i'll call it piece of m for males.

02:11 Now let's do the same for the females.

02:13 The females, we have 0 .45 as the percent, still a 95 % confidence interval.

02:20 And this time we'll have the 0 .45 times 1 minus, which is going to be 0 .55.

02:25 And then the sample size here was 522.

02:30 So we have, again, i'll do the minus sign first.

02:34 So 0 .45 minus 1 .96 times the square root of 0 .45 times 0 .55 divided by that 522.

02:47 And all of that's underneath the radical, just double checking.

02:50 And that gives me 40 .7%.

02:55 It's like 7 .3.

02:58 And go back again and change that so that i have that addition sign...