91Ó°ÊÓ

The table shows results from the 2014 General Social Survey on gender and whether one believes in an afterlife. $$ \begin{array}{lccc} \hline & \ {\text { Belief in Afterlife }} & \\ { 2 - 3 } \text { Gender } & \text { Yes } & \text { No } & \text { Total } \\\ \hline \text { Female } & 1026 & 207 & 1233 \\ \text { Male } & 757 & 252 & 1009 \\ \hline\end{array}$$ a. Denote the population proportion who believe in an afterlife by \(p_{1}\) for females and by \(p_{2}\) for males. Estimate \(p_{1}, p_{2},\) and \(\left(p_{1}-p_{2}\right)\) b. Find the standard error for the estimate of \(\left(p_{1}-p_{2}\right)\). Interpret. c. Construct a \(95 \%\) confidence interval for \(\left(p_{1}-p_{2}\right)\). Can you conclude which of \(p_{1}\) and \(p_{2}\) is larger? Explain. d. Suppose that, unknown to us, \(p_{1}=0.81\) and \(p_{2}=0.72 .\) Does the confidence interval in part c contain the parameter it is designed to estimate? Explain.

Step by step solution

01

Calculate Female Proportion (p1)

To find the population proportion for females who believe in an afterlife, divide the number of females who said 'Yes' by the total number of females surveyed. Therefore, \( p_1 = \frac{1026}{1233} \approx 0.832 \).

02

Calculate Male Proportion (p2)

For males, compute the proportion who believe in an afterlife by dividing the number of males who said 'Yes' by the total number of males surveyed. Thus, \( p_2 = \frac{757}{1009} \approx 0.750 \).

03

Calculate Difference in Proportions (p1 - p2)

Subtract the male proportion \( p_2 \) from the female proportion \( p_1 \) to find the difference in the proportions: \( p_1 - p_2 = 0.832 - 0.750 = 0.082 \).

04

Find Standard Error for Difference

The standard error for the difference in two proportions is calculated using the following formula: \[ SE = \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} } \] Substituting the given values: \[ SE = \sqrt{ \frac{0.832 \times 0.168}{1233} + \frac{0.750 \times 0.250}{1009} } \approx 0.021 \].

05

Construct 95% Confidence Interval

To construct a 95% confidence interval for \( p_1 - p_2 \), use the formula: \[ (p_1 - p_2) \pm Z\cdot SE \] where \( Z \approx 1.96 \) for a 95% confidence level. Therefore, the interval is: \[ 0.082 \pm 1.96 \times 0.021 \approx (0.041, 0.123) \]. Since the interval does not include zero and is entirely positive, \( p_1 \) is greater than \( p_2 \).

06

Check if Confidence Interval Contains True Difference

The given true proportions are \( p_1 = 0.81 \) and \( p_2 = 0.72 \), which means the true difference \( p_1 - p_2 = 0.09 \). This value falls within our calculated confidence interval of (0.041, 0.123), indicating that the interval does include the true parameter difference it is designed to estimate.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Difference in Proportions

When we talk about the difference in proportions, we're comparing two groups to see how common a certain characteristic is in each of them. In this case, we're looking at how many females versus males believe in an afterlife. To find these proportions, we divide the number of affirmative responses by the total surveyed in each gender group.

For example:

• The proportion of females believing in an afterlife, noted as \( p_1 \), is calculated as \( \frac{1026}{1233} \approx 0.832 \).
• For males, the proportion \( p_2 \) is \( \frac{757}{1009} \approx 0.750 \).

Subtracting these, we get the difference \( p_1 - p_2 = 0.832 - 0.750 = 0.082 \). This difference indicates that a higher proportion of females believe in an afterlife compared to males.

Understanding this difference helps us to see how these two groups compare in their beliefs.

Standard Error

Standard error is a measure of how much variation we might expect in the difference between two proportions. It gives us a sense of the uncertainty around the estimated difference in proportions. The formula for the standard error of the difference is:

• \[ SE = \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} } \]

Here, \( n_1 \) and \( n_2 \) represent the total number of surveyed females and males, respectively.

By plugging the information we have:

• Females: \( p_1 = 0.832 \), \( n_1 = 1233 \)
• Males: \( p_2 = 0.750 \), \( n_2 = 1009 \)

The standard error works out to be \( SE \approx 0.021 \). This value tells us the expected variability in the estimated difference between the two proportions. A smaller standard error suggests we have a more accurate estimate of the difference.

95% Confidence Level

Taking a deep dive into what a 95% confidence level means might help clarify the idea of confidence intervals. A confidence interval provides a range of values that are believed to contain the true difference in proportions, here between genders believing in an afterlife.

• To find this interval, we use the formula: \[ (p_1 - p_2) \pm Z \cdot SE \]
• For a 95% confidence level, \( Z \approx 1.96 \).
• Therefore, the interval becomes: \[ 0.082 \pm 1.96 \times 0.021 \approx (0.041, 0.123) \]

This confidence interval tells us that we are 95% confident the true difference in the proportions of belief in an afterlife between females and males lies between 0.041 and 0.123. Because this range does not include zero, it suggests that there is a statistically significant difference between the two groups, meaning that more females than males believe in an afterlife according to the survey results.