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Chapter 10: Problem 14

Fear of crime The elderly fear crime more than younger people, even though they are less likely to be victims of crime. One study recruited separate random samples of 56 black women and 63 black men over the age of 65 from Atlantic City, New Jersey. Of the women, 27 said they 鈥渇elt vulnerable鈥� to crime; 46 of the men said this.\(^{12}\) (a) Construct and interpret a 90% confidence interval for the difference in population proportions (men minus women). (b) Does your interval from part (a) give convincing evidence of a difference between the population proportions? Explain.

Short Answer

Expert verified
(0.0961, 0.4001); yes, there is a significant difference.

Step by step solution

01

Determine Sample Proportions

Calculate the sample proportions for both black women and black men. The sample proportion for black women \( \hat{p}_1 \) is calculated as \( \frac{27}{56} \approx 0.4821 \). The sample proportion for black men \( \hat{p}_2 \) is calculated as \( \frac{46}{63} \approx 0.7302 \).
02

Calculate Standard Error of Difference

Use the formula for the standard error (SE) of the difference in sample proportions: \[ SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \] Substituting the given values, \( n_1 = 56 \) and \( n_2 = 63 \), we find \[ SE = \sqrt{\frac{0.4821 \times 0.5179}{56} + \frac{0.7302 \times 0.2698}{63}} \approx 0.0924 \].
03

Find the Critical Value

Since we're constructing a 90% confidence interval, we find the critical value \( z^* \) for a confidence level of 90%. Using a standard normal distribution table, \( z^* \approx 1.645 \).
04

Calculate Confidence Interval

The confidence interval is given by: \[ (\hat{p}_2 - \hat{p}_1) \pm z^* \times SE \] Substituting the values we have: \( \hat{p}_2 - \hat{p}_1 = 0.7302 - 0.4821 = 0.2481 \). So, the interval is \[ 0.2481 \pm 1.645 \times 0.0924 \] which calculates to \[ 0.2481 \pm 0.1520 = (0.0961, 0.4001) \].
05

Interpret the Confidence Interval

The 90% confidence interval for the difference in population proportions of feelings of vulnerability to crime between men and women is \((0.0961, 0.4001)\). This interval suggests that the difference in the probability that a man feels vulnerable to crime compared to a woman is between 9.61% and 40.01%.
06

Conclusion on Evidence of Difference

Since the entire confidence interval for the difference in population proportions \((0.0961, 0.4001)\) is above zero, we have convincing evidence that the proportion of men who feel vulnerable to crime is significantly higher than that of women.

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Key Concepts

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

Population Proportions
Population proportions represent the ratio of individuals in a specific group who possess a certain characteristic. Understanding this concept is critical when comparing two different groups in a sample.
In our exercise, we are comparing two groups: black women and black men over the age of 65 in Atlantic City. Specifically, we want to know what portion of each group "feels vulnerable" to crime.
  • The sample proportion for black women, denoted by \( \hat{p}_1 \), is approximately 0.4821. This means about 48.21% of the sampled women felt vulnerable to crime.
  • The sample proportion for black men, denoted by \( \hat{p}_2 \), is approximately 0.7302, or about 73.02% of the men felt vulnerable.
By calculating these proportions, we lay the groundwork for further analysis like constructing confidence intervals.
Standard Error
The standard error (SE) measures the variability or dispersion of the sample estimate and helps us understand how much the estimated proportions might differ from the true population proportions. It is crucial in determining how precise our estimates are.
The formula used in our exercise is:
  • Standard Error for difference in proportions: \[ SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \]
Where \( n_1 = 56 \) and \( n_2 = 63 \), are the sample sizes for women and men, respectively.
Using this formula, we calculate the standard error to be approximately 0.0924. This value shows us the expected variability in the difference between the two population proportions if multiple samples were taken.
Critical Value
The critical value is a factor used to calculate the margin of error in a confidence interval. It is derived from the z-distribution, which assumes a normal distribution of the sample data.
For a 90% confidence level, a standard z-table or calculator is used to determine the critical value. In this instance, the critical value \( z^* \) is about 1.645.
The critical value tells us how many standard deviations away from the mean our bounds should be. It adjusts the width of the confidence interval, providing a range in which we can be fairly certain the true difference in population proportions falls. This concept helps to link the statistical theory with real-world data interpretation.
Interpretation of Results
Interpreting results is the final, yet crucial stage of statistical analysis. It involves understanding what the calculated confidence interval actually means in context.
Our confidence interval for the difference in population proportions is \((0.0961, 0.4001)\). This means we are 90% confident that the true difference in the proportion of men and women who "feel vulnerable" to crime lies between 9.61% and 40.01%.
  • The fact that this entire interval is above zero suggests that there is a significant difference between the two groups with regard to feeling vulnerable to crime.
  • This implies that, not only are men more likely to feel vulnerable, but we also have sufficient statistical evidence to back this claim based on the data.
Interpreting the results helps us decide if the hypothesis is statistically significant and provides insights that can influence further research or policy-making.

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Most popular questions from this chapter

Multiple choice: Select the best answer for Exercises 29 to 32. A sample survey interviews SRSs of 500 female college students and 550 male college students. Each student is asked whether he or she worked for pay last summer. In all, 410 of the women and 484 of the men say 鈥淵es.鈥� Exercises 29 to 31 are based on this survey. In an experiment to learn whether Substance M can help restore memory, the brains of 20 rats were treated to damage their memories. The rats were trained to run a maze. After a day, 10 rats (determined at random) were given M and 7 of them succeeded in the maze. Only 2 of the 10 control rats were successful. The two-sample z test for 鈥渘o difference鈥� against 鈥渁 significantly higher proportion of the M group succeeds鈥� (a) gives \(z=2.25, P<0.02\) (b) gives \(z=2.60, P<0.005\) (c) gives \(z=2.25, P<0.04\) but not \(<0.02\) (d) should not be used because the Random condition is violated. (e) should not be used because the Normal condition is violated.

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Multiple choice: Select the best answer for Exercises 29 to 32. A sample survey interviews SRSs of 500 female college students and 550 male college students. Each student is asked whether he or she worked for pay last summer. In all, 410 of the women and 484 of the men say 鈥淵es.鈥� Exercises 29 to 31 are based on this survey. The pooled sample proportion who worked last summer is about (a) \(\hat{p}_{\mathrm{C}}=1.70 . \quad(\mathrm{d}) \hat{p}_{\mathrm{C}}=0.85\) (b) \(\hat{p}_{\mathrm{C}}=0.89 . \quad\) (e) \(\hat{p}_{\mathrm{C}}=0.82\) (c) \(\hat{p}_{\mathrm{C}}=0.88\)

In Exercises 39 to 42, determine whether or not the conditions for using two- sample t procedures are met. Literacy rates Do males have higher average literacy rates than females in Islamic countries? The table below shows the percent of men and women at least 15 years old who were literate in 2008 in the major Islamic nations. (We omitted countries with populations of less than 3 million.) Data for a few nations, such as Afghanistan and Iraq, were not available.\(^{30}\) $$\begin{array}{lll}{\text { Country }} & {\text { Female percent }} & {\text { Male percent }} \\ {\text { Algeria }} & {66} & {94} \\ {\text { Bangladesh }} & {48} & {71} \\ {\text { Egypt }} & {58} & {88} \\ {\text { lran }} & {77}& {97} \\ {\text { Jordan }} & {87} & {99} \\ {\text { Kazakhstan }} & {100} & {100}\\\\{\text { Lebanon }} & {86} & {98} \\ {\text { Libya }} & {78} & {100}\\\\{\text { Libya }} & {78} & {100} \\ {\text { Malaysia }} & {90} & {98}\\\ {\text { Morocoo }} & {43}& {84} \\ {\text { Saudi Arabia }} & {79} & {98}\\\\{\text { Syria }} & {77} & {95} \\ {\text { Taijkistan }} & {100} & {100} \\ {\text { Tunisia }} & {69} & {97}\\\\{\text { Turkey }} & {81}& {99} \\ {\text { Uzbekistan }} & {96} & {99} \\ {\text { Yemen }} & {41} & {93}\end{array}$$

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