Question

For a number of years, nearly all Americans say that they would vote for a woman for president IF she were qualified, and IF she were from their own political party. But is America ready for a female president? A CBS/New York Times poll asked this question of a random sample of 1229 adults, with the following results: 19 $$\begin{array}{lcc} & \text { \% Responding "Yes } \\\& \text { Now } & 1999 \\\\\hline \text { Total } & 55 \% & 48 \% \\\\\text { Men } & 60 & 46 \\\\\text { Women } & 51 & 49 \\\\\text { Republicans } & 48 & 47 \\\\\text { Democrats } & 61 & 44 \\\\\text { Independents } & 55 & 54\end{array}$$ a. Construct a \(95 \%\) confidence interval for the proportion of all Americans who now believe that America is ready for a female president. b. If there were \(n_{1}=610\) men and \(n_{2}=619\) women in the sample, construct a \(95 \%\) confidence interval for the difference in the proportion of men and women who now believe that America is ready for a female president. Can you conclude that the proportion of men who now believe that America is ready for a female president is larger than the proportion of women? Explain. c. Look at the percentages of "yes" responses for Republicans, Democrats and Independents now compared to the percentages in \(1999 .\) Can you think of a reason why the percentage of Democrats might have changed so dramatically?

Short answer

a. Calculate the 95% confidence interval for the proportion of Americans who now believe that America is ready for a female president. The 95% confidence interval for the proportion of all Americans who believe America is ready for a female president is approximately: $$0.55 \pm 1.96\sqrt{\frac{0.55\times(1-0.55)}{1229}}$$ b. Calculate the 95% confidence interval for the difference in proportion between men and women, and determine if the proportion of men who now believe that America is ready for a female president is larger than women. The 95% confidence interval for the difference in proportions between men and women is approximately: $$\left(0.60-0.51\right) \pm 1.96\sqrt{\frac{0.60\times(1-0.60)}{610}+\frac{0.51\times(1-0.51)}{619}}$$ We need to check if the lower limit of the confidence interval calculated is greater than 0 to conclude if the proportion of men who now believe America is ready for a female president is larger than the proportion of women. c. Discuss the possible reasons for a significant change in the proportion of Democrats who believe that America is ready for a female president. Some possible reasons for the significant change in the proportion of Democrats who believe that America is ready for a female president might include increased presence and influence of female politicians within the Democratic party, a shift in the party's demographics, changes in political climate, and evolving societal attitudes towards women in leadership roles.

Step-by-step solution

Calculate the sample proportion for all Americans

Using the given data, we have 55% of the total sample that responded "yes." Therefore, the sample proportion, denoted as \(\hat{p}\), is: $$\hat{p} = \frac{\text{Number of Yes responses}}{\text{Total sample size}} = \frac{0.55\times1229}{1229}$$

Calculate the confidence interval for all Americans

Now, we will find the 95% confidence interval for the proportion of all Americans who believe that America is ready for a female president. The formula for the confidence interval is: $$\hat{p} \pm Z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$ Where \(Z_{\alpha/2}\) is the Z-score for a 95% confidence level, which equals 1.96. Substituting the values into the formula, we get: $$\hat{p} \pm 1.96\sqrt{\frac{0.55\times(1-0.55)}{1229}}$$

Calculate the sample proportions for men and women

Given that 60% of men and 51% of women responded "yes", and there were 610 men and 619 women, we can calculate the sample proportions of men and women as follows: $$\hat{p}_{1} = \frac{0.60\times610}{610}$$ $$\hat{p}_{2} = \frac{0.51\times619}{619}$$

Calculate the confidence interval for the difference in proportions between men and women

The formula for the confidence interval of the difference in proportions is: $$\left(\hat{p}_{1}-\hat{p}_{2}\right) \pm Z_{\alpha/2}\sqrt{\frac{\hat{p}_{1}(1-\hat{p}_{1})}{n_{1}}+\frac{\hat{p}_{2}(1-\hat{p}_{2})}{n_{2}}}$$ Now, substituting the values, we get: $$\left(0.60-0.51\right) \pm 1.96\sqrt{\frac{0.60\times(1-0.60)}{610}+\frac{0.51\times(1-0.51)}{619}}$$

Conclusion for can we conclude that the proportion of men who now believe that America is ready for a female president is larger than the proportion of women?

If the lower limit of the confidence interval calculated in step 4 is greater than 0, we can conclude that the proportion of men who now believe America is ready for a female president is larger than the proportion of women. Otherwise, we cannot make the conclusion.

Discuss the possible reasons for the change in Democrats' opinions

One possible reason why the percentage of Democrats who believe that America is ready for a female president changed so dramatically could be due to a more significant presence and influence of female politicians within the Democratic party. This might have made Democrats more open to the idea of having a female president. Additionally, there might have been a shift in demographics of the party, or changes in political climate and societal attitudes, contributing to the change in opinion.

Key concepts

Sample Proportion

Understanding a sample proportion involves examining a subset, or sample, of a larger population to make inferences about the entire group. It's like taking a snapshot of a select few to get an idea of the whole album. In statistical terms, the sample proportion, often denoted by \( \hat{p} \), represents the fraction of the sample with a particular attribute. For instance, if you're looking at how many students in a classroom wear glasses, and 10 out of 30 do, your sample proportion of students who wear glasses is \( \frac{10}{30} \), or approximately 0.33.

When we analyze the CBS/New York Times poll, we use the total number of 'yes' responses divided by the entire sample size to find the sample proportion. This simple ratio provides us with the necessary data to estimate what proportion of the full American population believes the country is ready for a female president. It acts as the foundation for constructing confidence intervals and testing hypotheses about the population's beliefs.

Statistical Significance

Statistical significance tells us whether the findings from our sample are likely reflective of the actual population. We want to know if what we're seeing isn't just a fluke but rather something meaningful that we can expect to find in the broader group. It's like deciding if a pattern in a sequence of coin tosses is random or if it points to the coin being rigged.

In context to our poll example, we're not just interested in knowing the sample proportions but also whether the observed difference or proportion is statistically significant. This means we're trying to determine if the findings in our sample—such as the percentages of different genders and political affiliations saying 'yes' to a female president—truly reflect the population's opinion or happened by chance. When we speak of confidence intervals and z-scores, we're essentially evaluating the statistical significance of our results.

Z-score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. Picture it as how many steps away you are from the middle of a bridge, where the middle is the average point. A z-score can indicate how uncommon or common a particular data point is within the data set.

For a 95% confidence interval, the z-score correlates to 1.96, which is related to the probability of a data point falling within two standard deviations of the mean on a standard normal distribution curve. In the case of our poll, we use this z-score to establish the margin of error around our sample proportion, which helps us express how sure we are that the true population proportion lies within a certain range around our sample proportion.

Difference in Proportions

When comparing two groups, it's not enough to know their individual proportions—we often want to evaluate the difference between them. This provides richer insights, especially when gauging the opinions or characteristics of two distinct segments. Imagine you have two baskets of fruit and you want to compare the proportion of apples in each. It's this difference that can tell you about the varying preferences or tendencies within the groups.

In reference to the poll data, calculating the difference in proportions between men and women's responses allows us to explore the gender dynamics in attitudes towards a female president. By constructing a confidence interval for this difference, we can assess whether the observed variance in opinions is statistically meaningful or could be due to random sampling variability. If our confidence interval for this difference does not include zero, it suggests a significant difference between men's and women's beliefs about the country being ready for a female president.