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Chapter 8: Problem 4

Explain the various alternatives for decreasing the width of a confidence interval. Which is the best alternative?

Short Answer

Expert verified
There are three alternatives for decreasing the width of a confidence interval. These include increasing the sample size, decreasing the variability in the population, and reducing the confidence level. The most effective and feasible method is usually to increase the sample size.

Step by step solution

01

Understanding Confidence Intervals

A Confidence Interval is a type of interval estimate in statistics that is used to estimate the range of values within which a population parameter is likely to lie with a certain level of confidence. The width of a confidence interval is determined by three factors: the sample size, the variability in the population, and the desired confidence level.
02

Increasing the Sample Size

One way to decrease the width of the confidence interval is by increasing the sample size. This is because as the sample size increases, the standard error decreases, reducing the margin of error and hence the width of the confidence interval. However, this may involve additional cost and time.
03

Decreasing the Confidence Level

Another method to reduce the width of the confidence interval is by decreasing the confidence level. A lower confidence level means that the interval will be narrower, but it also means that it is less likely to contain the true population parameter.
04

Decreasing Variability in the Population

Reducing the variability or standard deviation in the population also leads to a narrower confidence interval. However, this is often not practical as it is not something you can control.
05

Determining the Best Method

Increasing the sample size is often considered the best way to decrease the width of a confidence interval because it does not risk omitting the parameter of interest and within practical feasibility, unlike adjusting variability in population.

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Most popular questions from this chapter

On November 15,2006, carefair.com reported that \(40 \%\) of women aged 30 years and older would rather get Botox injections than spend a week in Paris. The survey consisted of 175 women in the specified age group. a. What is the point estimate of the corresponding population proportion? b. Construct a \(98 \%\) confidence interval for the proportion of all women aged 30 years and older who would rather get Botox injections than spend a week in Paris. What is the margin of error for this estimate?

Calculating a confidence interval for the proportion requires a minimum sample size. Calculate a confidence interval, using any confidence level of \(90 \%\) or higher, for the population proportion for each of the following. a. \(n=200 \quad\) and \(\quad \hat{p}=.01\) b. \(n=160 \quad\) and \(\quad \hat{p}=.9875\) Explain why these confidence intervals reveal a problem when the conditions for using the normal approximation do not hold.

A random sample of 20 acres gave a mean yield of wheat equal to \(41.2\) bushels per acre with a standard deviation of 3 bushels. Assuming that the yield of wheat per acre is normally distributed, construct a \(90 \%\) confidence interval for the population mean \(\mu .\)

A company that produces 8 -ounce low-fat yogurt cups wanted to estimate the mean number of calories for such cups. A random sample of 10 such cups produced the following numbers of calories. \(\begin{array}{lllllllll}147 & 159 & 153 & 146 & 144 & 148 & 163 & 153 & 143 & 158\end{array}\) Construct a \(99 \%\) confidence interval for the population mean. Assume that the numbers of calories for such cups of yogurt produced by this company have an approximately normal distribution.

In June 2008 , SBRI Public Affairs conducted a telephone poll of 1004 adult Americans aged 18 and older. One of the questions asked was, "In the past year, was there ever a time when you ...?" Respondents could choose more than one of the answers mentioned. Of the respondents, \(64 \%\) said "cut back on vacations or entertainment because of their cost," \(37 \%\) said "failed to pay a bill on time," and \(25 \%\) said "have not gone to a doctor because of the cost." (Source: http://www.srbi.com/AmericansConcernEconomic.html.) Using these results, find a \(95 \%\) confidence interval for the corresponding population percentage for each answer. Write a one-page report to present these results to a group of college students who have not taken statistics. Your report should answer questions such as: (1) What is a confidence interval? (2) Why is a range of values more informative than a single percentage? (3) What does \(95 \%\) confidence mean in this context? (4) What assumptions, if any, are you making when you construct each confidence interval?
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