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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

A random sample of students who took the SAT college entrance examination twice found that 427 of the respondents had paid for coaching courses and that the remaining 2733 had not. \({ }^{14}\) Construct and interpret a \(99 \%\) confidence interval for the proportion of coaching among students who retake the SAT.

Most people can roll their tongues, but many can't. The ability to roll the tongue is genetically determined. Suppose we are interested in determining what proportion of students can roll their tongues. We test a simple random sample of 400 students and find that 317 can roll their tongues. The margin of error for a \(95 \%\) confidence interval for the true proportion of tongue rollers among students is closest to (a) 0.0008 . (c) 0.03 . (c) 0.05 . (b) 0.02 (d) 0.04

The Trial Urban District Assessment (TUDA) is a government-sponsored study of student achievement in large urban school districts. TUDA gives a reading test scored from 0 to \(500 . \mathrm{A}\) score of 243 is a "basic" reading level and a score of 281 is "proficient." Scores for a random sample of 1470 eighth- graders in Atlanta had \(\bar{x}=240\) with standard deviation \(42.17 .^{24}\) (a) Calculate and interpret a \(99 \%\) confidence interval for the mean score of all Atlanta eighth-graders. (b) Based on your interval from part (a), is there good evidence that the mean for all Atlanta eighth-graders is less than the basic level? Explain.

High school students who take the SAT Math exam a second time generally score higher than on their first try. Past data suggest that the score increase has a standard deviation of about 50 points. How large a sample of high school students would be needed to estimate the mean change in SAT score to within 2 points with \(95 \%\) confidence? Show your work.

A quality control inspector will measure the salt content (in milligrams) in a random sample of bags of potato chips from an hour of production. Which of the following would result in the smallest margin of error in estimating the mean salt content \(\mu ?\) (a) \(90 \%\) confidence; \(n=25\) (b) \(90 \%\) confidence; \(n=50\) (c) \(95 \%\) confidence; \(n=25\) (d) \(95 \%\) confidence; \(n=50\) (e) \(n=100\) at any confidence level
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