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Chapter 8: Problem 34

You are interested in estimating the mean commuting time from home to school for all commuter students at your school. Briefly explain the procedure you will follow to conduct this study. Collect the required data from a sample of 30 or more such students and then estimate the population mean at a \(99 \%\) confidence level. Assume that the population standard deviation for such times is \(5.5\) minutes.

Short Answer

Expert verified
The procedure includes selecting a representative sample of 30 or more students, recording their commuting times, calculating the sample mean and estimating the population mean with a 99% confidence interval using the formula: \(\overline{X} \pm (2.576 \cdot \frac{5.5}{\sqrt{n}})\). The exact figures would depend on the collected sample data.

Step by step solution

01

Explaining the Procedure

The first step in conducting this study is to select a sample of commuter students. The size of the sample has been fixed at a minimum of 30 students. It should be collected in a random manner to ensure a representative sample of all commuter students. The commuting time from home to school for each student in the sample needs to be recorded.
02

Compute Sample Mean

Once the data has been collected, calculate the average commuting time of the sample. This can be done by adding up all the individual commuting times and dividing by the total number of students in the sample which is \(n\). Let this sample mean be denoted by \(\overline{X}\).
03

Estimate Population Mean

Estimate the population mean using a 99% confidence interval. We know that the population standard deviation \(\sigma\) is given as \(5.5\) minutes. For a 99% confidence level, the z-score (z) from the standard deviation table is \(2.576\). A 99% confidence interval for the population mean is given by the formula \(\overline{X} \pm (z \cdot \frac{\sigma}{\sqrt{n}})\). From here, proceed to substitute \(\sigma\), \(n\), and \(z\) into the formula to get the confidence interval.

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Most popular questions from this chapter

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.

A hospital administration wants to estimate the mean time spent by patients waiting for treatment at the emergency room. The waiting times (in minutes) recorded for a random sample of 35 such patients are given below. \(\begin{array}{lrrrrrr}30 & 7 & 68 & 76 & 47 & 60 & 51 \\ 64 & 25 & 35 & 29 & 30 & 35 & 62 \\ 96 & 104 & 58 & 32 & 32 & 102 & 27 \\ 45 & 11 & 64 & 62 & 72 & 39 & 92 \\ 84 & 47 & 12 & 33 & 55 & 84 & 36\end{array}\) Construct a \(99 \%\) confidence interval for the corresponding population mean. Use the \(t\) distribution.

Check if the sample size is large enough to use the normal distribution to make a confidence interval for \(p\) for each of the following cases. a. \(n=80\) and \(\hat{p}=.85\) b. \(n=110\) and \(\hat{p}=.98\) c. \(n=35\) and \(\hat{p}=.40\) d. \(n=200\) and \(\hat{p}=.08\)

KidPix Entertainment 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 labor-hours 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?

What is the point estimator of the population mean, \(\mu ?\) How would you calculate the margin of error for an estimate of \(\mu\) ?
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