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Chapter 8: Problem 15

Statistics class in Exercise 1 also asked an SRS of 20 boys at their school how many pairs of shoes they have. A \(95 \%\) confidence interval for the difference in the population means (girls - boys) is 10.9 to \(26.5 .\) Interpret the confidence interval and the confidence level.

Short Answer

Expert verified
We are 95% confident that girls have 10.9 to 26.5 more pairs of shoes than boys, meaning the true difference in means lies within this range.

Step by step solution

01

Understanding the Confidence Interval

The confidence interval provided is from 10.9 to 26.5. This means that, statistically speaking, we are 95% confident that the true difference in the average number of pairs of shoes owned by girls and boys lies within this interval. The interval suggests that, on average, girls own between 10.9 and 26.5 more pairs of shoes than boys.
02

Interpreting the Confidence Level

The 95% confidence level indicates that if we were to take many samples and construct a confidence interval from each one, about 95% of those intervals would contain the true difference in population means. It reflects a high level of certainty consistent with the interval specified, showing reliability in this estimate.

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Key Concepts

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

Population Means
Population means are essentially the average values of a particular characteristic for an entire group or population. In the context of our exercise, we're comparing the mean number of pairs of shoes owned by girls versus boys. The population mean is a theoretical value representing what the average would be if we surveyed every single member of the population, girls or boys in this case.
Simplifying this further:
  • The population mean provides a central point in understanding the characteristic across everyone in that population group.
  • It allows us to compare these averages between two distinct populations, such as girls and boys, to discern meaningful differences.
In our problem, we have a 95% confidence interval for the difference in population means, indicating that girls, on average, tend to own more pairs of shoes than boys within the specified range.
Confidence Level
The confidence level is a critical concept used to quantify the degree of certainty in statistics. In our exercise, a 95% confidence level was used, which is quite common. Here's how the concept plays out:
  • A 95% confidence level means we are 95% certain that the computed interval contains the true difference between the population means of two groups.
  • It's about the reliability of our estimation process. If we were to repeat the sampling process 100 times, we expect 95 of those confidence intervals to contain the true difference between the populations.
This concept helps measure how sure we are about our predictions, offering a balance between precision and practicality. In our exercise, it implies robust confidence that the true difference in the average number of shoe pairs owned by girls and boys lies between 10.9 and 26.5 pairs, though there's still a 5% chance that it does not.
Simple Random Sample
A simple random sample (SRS) is a fundamental sampling method in statistics. It refers to the unbiased selection of samples where each member of the population has an equal chance of being chosen. In our exercise, an SRS of 20 boys was used to gather data. Here's why this method is vital:
  • Ensures each sampled individual is randomly chosen, minimizing selection bias.
  • Increases the likelihood that the sample accurately reflects the larger population.
  • Simplifies the process of data collection while maintaining the integrity of the sample.
By employing an SRS, the statistics class could confidently characterize the shoe-owning habits of boys, knowing their sample is representative of the broader population of boys at the school. This methodology helps underlie the reliability of the entire statistical analysis.

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Most popular questions from this chapter

Determine whether we can safely use a \(t^{*}\) critical value to calculate a confidence interval for the population mean in each of the following settings. (a) We collect data from a random sample of adult residents in a state. Our goal is to estimate the overall percent of adults in the state who are college graduates. (b) The coach of a college men's basketball team records the resting heart rates of the 15 team members. We use these data to construct a confidence interval for the mean resting heart rate of all male students at this college. (c) Do teens text more than they call? To find out, an \(\mathrm{AP}^{8}\) Statistics class at a large high school collected data on the number of text messages and calls sent or received by each of 25 randomly selected students. The Fathom boxplot below displays the difference (texts - calls) for each student.

In a recent National Survey of Drug Use and Health, 2312 of 5914 randomly selected full-time U.S. college students were classified as binge drinkers. \({ }^{13}\) (a) Calculate and interpret a \(99 \%\) confidence interval for the population proportion \(p\) that are binge drinkers. (b) A newspaper article claims that \(45 \%\) of full-time U.S. college students are binge drinkers. Use your result from part (a) to comment on this claim.

Check whether each of the conditions is met for calculating a confidence interval for the population proportion \(\bar{p}\). Latoya wants to estimate what proportion of the seniors at her boarding high school like the cafeteria food. She interviews an SRS of 50 of the 175 seniors living in the dormitory. She finds that 14 think the cafeteria food is good.

A New York Times/CBS News Poll asked a random sample of U.S. adults the question, "Do you favor an amendment to the Constitution that would permit organized prayer in public schools?" Based on this poll, the \(95 \%\) confidence interval for the population proportion who favor such an amendment is (0.63,0.69) (a) Interpret the confidence interval. (b) What is the point estimate that was used to create the interval? What is the margin of error? (c) Based on this poll, a reporter claims that more than two-thirds of U.S. adults favor such an amendment. Use the confidence interval to evaluate this claim.

Have efforts to promote equality for women gone far enough in the United States? A poll on this issue by the cable network MSNBC contacted 1019 adults. A newspaper article about the poll said, "Results have a margin of sampling error of plus or minus 3 percentage points." (a) The news article said that \(65 \%\) of men, but only \(43 \%\) of women, think that efforts to promote equality have gone far enough. Explain why we do not have enough information to give confidence intervals for men and women separately. (b) Would a \(95 \%\) confidence interval for women alone have a margin of error less than \(0.03,\) about equal to \(0.03,\) or greater than 0.03 ? Why? (You see that the news article's statement about the margin of error for poll results is a bit misleading.)
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