7. Sampling Distributions & Confidence Intervals: Mean

What does the 95% represent in a 95% confidence interval?

Reading Rates (See Problem 22 in Section 10.3.) Michael Sullivan, son of the author, decided to enroll in a reading course that allegedly increases reading speed and comprehension. Prior to enrolling in the course, Michael read 198 words per minute (wpm). The following data represent the words per minute read for 10 different passages after the course.

c. Generate 5000 independent bootstrap samples of size n=10 with replacement. For each bootstrap sample, determine the sample mean. That is, build a null model.

"Time Spent in the Drive-Thru The quality-control manager of a Long John Silver’s restaurant wants to analyze the length of time that a car spends at the drive-thru window waiting for an order. It is determined that the mean time spent at the window is 59.3 seconds with a standard deviation of 13.1 seconds. The distribution of time at the window is skewed right (data based on information provided by Danica Williams, student at Joliet Junior College).

a. To obtain probabilities regarding a sample mean using the normal model, what size sample is required?"

"Insect Fragments The Food and Drug Administration sets Food Defect Action Levels (FDALs) for some of the various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL for insect filth in peanut butter is 3 insect fragments (larvae, eggs, body parts, and so on) per 10 grams. A random sample of 50 ten-gram portions of peanut butter is obtained and results in a sample mean of x_bar=3.6 insect, fragments per ten-gram portion.

a. Why is the sampling distribution of x_bar approximately normal?"

"Burger King’s Drive-Thru Suppose that cars arrive at Burger King’s drive-thru at the rate of 20 cars every hour between 12:00 noon and 1:00 P.M. A random sample of 40 one-hour time periods between 12:00 noon and 1:00 P.M. is selected and has 22.1 as the mean number of cars arriving.

a. Why is the sampling distribution of x_bar approximately normal?"

"Watching Television The amount of time Americans spend watching television is closely monitored by firms such as AC Nielsen because this helps determine advertising pricing for commercials.

d. Determine the likelihood of obtaining a sample mean of 1.89 hours or less from a population whose mean is presumed to be 2.35 hours. Based on the result obtained, do you think avid Internet users watch less television?"

"Sleepy Suppose you want to study the number of hours of sleep you get each evening. To do so, you look at the calendar and randomly select 10 days out of the next 300 days and record the number of hours you sleep.

d. Is the mean number of hours computed in part (c) a random variable? Why? If it is a random variable, what is the source of variation"

Under what conditions is the sampling distribution of x̄ normal?

"The charge life of a certain lithium ion battery for camcorders is normally distributed, with mean 90 minutes and standard deviation 35 minutes.

b. Describe the sampling distribution of x̄, the sample mean charge life for a random sample of 10 such batteries."

The notation tα is the t-value such that the area under the t-distribution to the right of tα is .

Long Life? In a survey of 35 adult Americans, it was found that the mean age (in years) that people would like to live to is 87.9 with a standard deviation of 15.5. An analysis of the raw data indicates the distribution is skewed left.

a. Explain why a large sample size is necessary to construct a confidence interval for the mean age that people would like to live.