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Chapter 8: Problem 5

Briefly explain how the width of a confidence interval decreases with an increase in the sample size. Give an example.

Short Answer

Expert verified
The width of a confidence interval decreases with an increase in sample size because larger samples provide a better representation of the population, leading to more precise estimations. An example is measuring people's heights in a city; as the number of people measured increases, the confidence interval for the average height narrows.

Step by step solution

01

Define Confidence Interval

A confidence interval can be defined as a range of values, derived from statistical calculati ons, in which we believe the true population value lies. The width of the confidence interval suggests the precision of the estimation.
02

Explain Sample Size Impact on Confidence Interval Width

As the sample size increases, statistical theory suggests that the width (precision) of our confidence intervals should decrease. This is because larger samples are more likely to better represent the full population, leading to more precise (narrower) intervals.
03

Provide an Example

Consider a study estimating the average height of people in a city. If we measure 10 people, our confidence interval might be quite wide due to the small sample. However, if we measure 1,000 people, our increased sample size will likely give us a more precise (narrower) confidence interval for the average height.

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Most popular questions from this chapter

Determine the most conservative sample size for the estimation of the population proportion for the following. a. \(E=.025\), confidence level \(=95 \%\) b. \(E=.05, \quad\) confidence level \(=90 \%\) c. \(E=.015\), confidence level \(=99 \%\)

An entertainment company is in the planning stages of producing a new computer-animated movie for national release, so they need to determine the production time (labor-hours necessary) to produce the movie. The mean production time for a random sample of 14 big-screen computer-animated movies is found to be 53,550 labor-hours. Suppose that the population standard deviation is known to be 7462 laborhours and the distribution of production times is normal. a. Construct a \(98 \%\) confidence interval for the mean production time to produce a big-screen computer-animated movie. b. Explain why we need to make the confidence interval. Why is it not correct to say that the average production time needed to produce all big-screen computer-animated movies is 53,550 labor-hours?

The high price of medicines is a source of major expense for those seniors in the United States who have to pay for these medicines themselves. A random sample of 2000 seniors who pay for their medicines showed that they spent an average of \(\$ 4600\) last year on medicines with a standard deviation of \(\$ 800\). Make a \(98 \%\) confidence interval for the corresponding population mean.

a. A sample of 1100 observations taken from a population produced a sample proportion of .32. Make a \(90 \%\) confidence interval for \(p\). b. Another sample of 1100 observations taken from the same population produced a sample proportion of .36. Make a \(90 \%\) confidence interval for \(p\). c. A third sample of 1100 observations taken from the same population produced a sample proportion of .30. Make a \(90 \%\) confidence interval for \(p\). d. The true population proportion for this population is \(.34 .\) Which of the confidence intervals constructed in parts a through c cover this population proportion and which do not?

A sample of 20 managers was taken, and they were asked whether or not they usually take work home. The responses of these managers are given below, where yes indicates they usually take work home and no means they do not. \(\begin{array}{lllllllll}\text { Yes } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { No } \\ \text { Yes } & \text { Yes } & \text { No } & \text { Yes } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { No } & \text { Yes }\end{array}\) Make a \(99 \%\) confidence interval for the percentage of all managers who take work home.
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