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Chapter 6: Problem 144

Gender and Gun Control A survey reported in Time magazine included the question "Do you favor a federal law requiring a 15 day waiting period to purchase a gun?" Results from a random sample of US citizens showed that 318 of the 520 men who were surveyed supported this proposed law while 379 of the 460 women sampled said "yes". Use this information to find and interpret a \(90 \%\) confidence interval for the difference in the proportions of men and women who agree with this proposed law.

Short Answer

Expert verified
The 90% confidence interval for the difference in proportions of men and women supporting the 15-day waiting period law to purchase a gun would be calculated using the above steps. The interval would indicate the range in which we're 90% confident that the true difference in population proportions lies.

Step by step solution

01

Calculate the proportions

To begin with, we need to calculate the proportions of males and females that support the law. This is done by dividing the number of people that agree with the proposition by the total number. Let's denote \(p_1\) as proportion of males supporting the law and \(p_2\) as proportion of females supporting the law.\n\nCompute \(p_1 = \frac{318}{520}\) and \(p_2 = \frac{379}{460}\).
02

Compute the Difference in Proportions

The difference in proportions is computed as \(\delta = p_1 - p_2\). Calculate \(\delta\).
03

Calculate the Standard Error

The standard error (SE) for the difference in proportions is calculated using the formula \n\n\[SE = \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }\] \n\nwhere \(n_1\) and \(n_2\) are the number of males and females surveyed. Compute the SE.
04

Use the Z-table

For our 90% confidence interval, the corresponding z-value is 1.645. Look up this value in a z table or use software to find it.
05

Calculate the Confidence Interval

The confidence interval is calculated as \(\delta \pm z * SE\). Compute the lower and upper bounds of the interval.

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Key Concepts

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

Proportions
In statistics, a proportion is a way to describe a part of a whole. In surveys, it helps express how many people in a group support a certain opinion or behavior. Considering the survey data from the original exercise, we are focusing on two groups: men and women.
For the surveyed men, the proportion supporting federal law is calculated as follows:
  • Number of men supporting the law = 318
  • Total number of men surveyed = 520
  • Proportion of men supporting = \( p_1 = \frac{318}{520} \approx 0.6115 \)
Similarly, we calculate the proportion for women:
  • Number of women supporting the law = 379
  • Total number of women surveyed = 460
  • Proportion of women supporting = \( p_2 = \frac{379}{460} \approx 0.8239 \)
These proportions give us an insight into each group's stance on the law, allowing us to compare them effectively.
Standard Error
The standard error (SE) measures how much variation we can expect in sample statistics, such as the difference of proportions. When dealing with proportions from survey data, it's crucial to understand how confident we can be in the results.
For the difference in proportions between men and women who support the law, the standard error is calculated using the formula: \[SE = \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }\] where:
  • \( p_1 \) and \( p_2 \) are the proportions of men and women supporting the law, respectively.
  • \( n_1 \) and \( n_2 \) are the sample sizes for men and women.
By calculating the standard error, we can determine how much the difference between these sample proportions may fluctuate across different samples. This is crucial for constructing a meaningful confidence interval, as a smaller SE results in a more precise interval.
Survey Analysis
Survey analysis involves interpreting survey data to make conclusions about a population. The original exercise requires us to create a confidence interval to see how much the support for gun control laws might differ between men and women in the larger population.
A 90% confidence interval uses a critical z-value of 1.645, reflecting the 10% risk (5% on each side) of error. We calculate this interval from our sample proportions and standard error. The formula is: \[CI = \delta \pm z \times SE\]where \( \delta \) is the difference in the sample proportions.
Creating a confidence interval allows the survey analysis to express a range in which the true difference of population proportions likely falls. This helps policymakers or researchers understand public opinion with a quantifiable level of certainty, guiding them in making informed decisions based on potential demographic differences.

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Most popular questions from this chapter

When we want \(95 \%\) confidence and use the conservative estimate of \(p=0.5,\) we can use the simple formula \(n=1 /(M E)^{2}\) to estimate the sample size needed for a given margin of error ME. In Exercises 6.40 to 6.43, use this formula to determine the sample size needed for the given margin of error. A margin of error of 0.02 .

When we want \(95 \%\) confidence and use the conservative estimate of \(p=0.5,\) we can use the simple formula \(n=1 /(M E)^{2}\) to estimate the sample size needed for a given margin of error ME. In Exercises 6.40 to 6.43, use this formula to determine the sample size needed for the given margin of error. A margin of error of 0.01

Use StatKey or other technology to generate a bootstrap distribution of sample proportions and find the standard error for that distribution. Compare the result to the standard error given by the Central Limit Theorem, using the sample proportion as an estimate of the population proportion \(p\). Proportion of survey respondents who say exercise is important, with \(n=1000\) and \(\hat{p}=0.753\)

A data collection method is described to investigate a difference in means. In each case, determine which data analysis method is more appropriate: paired data difference in means or difference in means with two separate groups. To study the effect of sitting with a laptop computer on one's lap on scrotal temperature, 29 men have their scrotal temperature tested before and then after sitting with a laptop for one hour.

Use the t-distribution and the given sample results to complete the test of the given hypotheses. Assume the results come from random samples, and if the sample sizes are small, assume the underlying distributions are relatively normal. Test \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1}>\mu_{2}\) using the sample results \(\bar{x}_{1}=56, s_{1}=8.2\) with \(n_{1}=30\) and \(\bar{x}_{2}=51, s_{2}=6.9\) with \(n_{2}=40\).
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