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Chapter 9: Problem 33

Confidence Interval Changes State whether each of the following changes would make a confidence interval wider or narrower. (Assume that nothing else changes.) a. Changing from a \(90 \%\) confidence level to a \(99 \%\) confidence level b. Changing from a sample size of 30 to a sample size of 200 c. Changing from a standard deviation of 20 pounds to a standard deviation of 25 pounds

Short Answer

Expert verified
a. The confidence interval would become wider. b. The confidence interval would become narrower. c. The confidence interval would become wider.

Step by step solution

01

Response to Situation A

When the confidence level increases from \(90\%\) to \(99\%\), the Z score also increases (as you are demanding more certainty about the sample mean being representative of the population mean). Hence, the width of the confidence interval would increase. Therefore, the confidence interval becomes wider.
02

Response to Situation B

When the sample size increases from 30 to 200, the denominator in the term \(\frac{\sigma}{\sqrt{n}}\) increases. As a result, the value of \(Z \cdot \frac{\sigma}{\sqrt{n}}\) decreases, hence reducing the width of the confidence interval. Therefore, the confidence interval becomes narrower.
03

Response to Situation C

When the standard deviation increases from 20 to 25, the numerator in the term \(\frac{\sigma}{\sqrt{n}}\) increases. As a result, the value of \(Z \cdot \frac{\sigma}{\sqrt{n}}\) increases and hence the width of the confidence interval increases. Therefore, the confidence interval becomes wider.

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Key Concepts

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

Confidence Level
The confidence level is a crucial concept in statistics, reflecting how certain we are that a confidence interval contains the true population parameter. The confidence level is often expressed as a percentage, such as 90% or 99%.
  • When you choose a higher confidence level, such as moving from 90% to 99%, you are aiming for greater certainty. This means you are saying, "I want to be more confident that the interval captures the true parameter."
  • As a result, the interval becomes wider to accommodate this increased certainty. It acts like casting a wider net to ensure you catch the correct parameter.
Understanding this can help you decide what level of confidence is appropriate based on the trade-off between certainty and precision. A higher confidence level means more certainty, but it also means a less precise estimate.
Sample Size
Sample size directly impacts the width of a confidence interval. A larger sample size can lead to more precise estimates.
  • When you increase the sample size, like going from 30 to 200, the confidence interval tends to become narrower. This is because you're getting more data, which generally leads to a more accurate estimate of the population mean.
  • The mathematical reasoning behind this involves the formula for the standard error: \( \frac{\sigma}{\sqrt{n}} \), where \( n \) is the sample size.
The denominator \( \sqrt{n} \) increases with larger sample sizes, reducing the overall value of the standard error, making the confidence interval narrower. Hence, larger samples provide more precise insights into the population.
Standard Deviation
Standard deviation measures the amount of variability or dispersion in a dataset. It directly influences the width of a confidence interval.
  • If the standard deviation increases, like from 20 pounds to 25 pounds, the confidence interval width increases. This is because greater variability means there's more uncertainty about the sample mean as an estimate of the population mean.
  • The formula \( Z \cdot \frac{\sigma}{\sqrt{n}} \) shows that an increase in \( \sigma \) (standard deviation) results in a larger product, thereby extending the interval.
In simple terms, if the data points are more spread out, we need to make the interval wider to maintain the same confidence level. Hence, a higher standard deviation reflects more variability and wider confidence intervals.

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Most popular questions from this chapter

Travel Time to School A random sample of \(50 \mathrm{1} 2\) th-grade students was asked how long it took to get to school. The sample mean was \(16.2\) minutes, and the sample standard deviation was \(12.3\) minutes. (Source: AMSTAT Census at School) a. Find a \(95 \%\) confidence interval for the population mean time it takes 12 th-grade students to get to school. b. Would a \(90 \%\) confidence interval based on this sample data be wider or narrower than the \(95 \%\) confidence interval? Explain. Check your answer by constructing a \(90 \%\) confidence interval and comparing this width of the interval with the width of the \(95 \%\) confidence interval you found in part a.

Presidents' Ages at Inauguration A \(95 \%\) confidence interval for the ages of the first six presidents at their inaugurations is \((56.2,59.5) .\) Either interpret the interval or explain why it should not be interpreted.

Pulse and Gender: Cl Using data from NHANES, we looked at the pulse rate for nearly 800 people to see whether it is plausible that men and women have the same population mean. NHANES data are random and independent. Minitab output follows.

Retirement Income Several times during the year, the U.S. Census Bureau takes random samples from the population. One such survey is the American Community Survey. The most recent such survey, based on a large (several thousand) sample of randomly selected households, estimates the mean retirement income in the United States to be \(\$ 21,201\) per year. Suppose we were to make a histogram of all of the retirement incomes from this sample. Would the histogram be a display of the population distribution, the distribution of a sample, or the sampling distribution of means?

Deflated Footballs? Patriots In the 2015 AFC Championship game, there was a charge the New England Patriots deflated their footballs for an advantage. The balls should be inflated to between \(12.5\) and \(13.5\) pounds per square inch. The measurements were \(11.50,10.85,11.15,10.70,11.10,11.60,11.85,11.10,10.95\) \(10.50\), and \(10.90\) psi (pounds per square inch). (Source: http:// online.wsj.com/public/resources/documents/Deflategate.pdf) a. Test the hypothesis that the population mean is less than \(12.5\) psi using a significance level of \(0.05\). State clearly whether the Patriots' balls are deflated or not. Assume the conditions for a hypothesis test are satisfied. b. Use the data to construct a \(95 \%\) confidence interval for the mean psi for the Patriots' footballs. How does this confidence interval support your conclusion in part a?
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