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Chapter 8: Problem 9

How will you interpret a \(99 \%\) confidence interval for \(\mu\) ? Explain.

Short Answer

Expert verified
A 99% confidence interval for the population mean, 渭, suggests that we are 99% confident that the confidence interval contains the true population mean. This indicates if we repeatedly took samples and constructed confidence intervals in the same way, 99% of them would contain the actual population mean, 渭. However, it is not ensured that any specific interval contains 渭.

Step by step solution

01

Understanding the Concept of Confidence Interval

A confidence interval can be defined as a type of estimate computed from the statistics of the observed data. This proposes a range of plausible values for an unknown parameter (in this case, the population mean, denoted as 渭). The confidence level, here 99%, defines the probability that the interval will contain the true population parameter, 渭, when you draw a random sample from the population.
02

Confidence Interval Interpretation

The correct interpretation of a 99% confidence interval is: 'We are 99% confident that the true population mean, 渭, lies within the confidence interval.' This means, if we were to draw random samples from the population and calculated an interval estimate each time, then 99% of these intervals would contain the true mean of the population, 渭.
03

Clarifying Misconception about Confidence Interval

It's crucial to understand that a 99% confidence level doesn鈥檛 imply that the population parameter will fall into the confidence interval 99% of the time. Instead, it means that 99% of randomly drawn samples will result in confidence intervals that cover the actual population parameter, 渭.

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Most popular questions from this chapter

Briefly explain the similarities and the differences between the standard normal distribution and the \(t\) distribution.

The principal of a large high school is concerned about the amount of time that his students spend on jobs to pay for their cars, to buy clothes, and so on. He would like to estimate the mean number of hours worked per week by these students. He knows that the standard deviation of the times spent per week on such jobs by all students is \(2.5\) hours. What sample size should he choose so that the estimate is within \(.75\) hour of the population mean? The principal wants to use a \(98 \%\) confidence level.

Determine the sample size for the estimation of the population proportion for the following, where \(\hat{p}\) is the sample proportion based on a preliminary sample. a. \(E=.025, \hat{p}=.16, \quad\) confidence level \(=99 \%\) b. \(E=.05, \quad \hat{p}=.85, \quad\) confidence level \(=95 \%\) c. \(E=.015, \quad \hat{p}=.97, \quad\) confidence level \(=90 \%\)

You are interested in estimating the mean age of cars owned by all people in the United States. Briefly explain the procedure you will follow to conduct this study. Collect the required data on a sample of 30 or more cars and then estimate the population mean at a \(95 \%\) confidence level. Assume that the population standard deviation is \(2.4\) years,

A researcher wanted to know the percentage of judges who are in favor of the death penalty. He took a random sample of 15 judges and asked them whether or not they favor the death penalty. The responses of these iudges are given here. \(\begin{array}{lllllll}\text { Yes } & \text { No } & \text { Yes } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { Yes } \\ \text { Yes } & \text { No } & \text { Yes } & \text { Yes } & \text { Yes } & \text { No } & \text { Yes }\end{array}\) a. What is the point estimate of the population proportion? b. Make a \(95 \%\) confidence interval for the percentage of all judges who are in favor of the death penalty.
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