7. Sampling Distributions & Confidence Intervals: Mean

The procedure for constructing a t-interval is robust. Explain what this means.

c. Find the t-value such that the area left of the t-value is 0.01 with 18 degrees of freedom. (Hint: Use symmetry.)

d. Find the critical t-value that corresponds to 90% confidence. Assume 20 degrees of freedom.

Which Is More Likely? Assume that the fertility rates in Exercise 32 are normally distributed. Are you more likely to randomly select a state with a fertility rate of less than 65 or to randomly select a sample of 15 states in which the mean of the state fertility rates is less than 65? Explain.

"Upper Leg Length The upper leg length of 20- to 29-year-old males is approximately normal with a mean length of 43.7 cm and a standard deviation of 4.2 cm.

Source: “Anthropometric Reference Data for Children and Adults: U.S. Population, 1999–2002”; Volume 361, July 7, 2005.

e. A random sample of 15 males who are 20–29 years old results in a mean upper leg length of 46 cm. Do you find this result unusual? Why?"

True or False: To cut the standard error of the mean in half, the sample size must be doubled.

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

"Reading Rates The reading speed of second-grade students is approximately normal, with a mean of 90 words per minute (wpm) and a standard deviation of 10 wpm.

d. What effect does increasing the sample size have on the probability? Provide an explanation for this result."

"Answer the following questions for the sampling distribution of the sample mean shown to the right.

b. What is the value of σx̄?"

Choosing a Distribution In Exercises 35–40, use the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

Body Mass Index In a random sample of 50 people, the mean body mass index (BMI) was 27.7 and the standard deviation was 6.12.

In Exercises 55–60, find the indicated probabilities and interpret the results.

The mean ACT composite score in a recent year is 20.7. A random sample of 36 ACT composite scores is selected. What is the probability that the mean score for the sample is (a) less than 22, (b) greater than 23, and (c) between 20 and 21.5? Assume sigma=5.9.

Suppose a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x̄ has mean μx̄ = _________ and standard deviation σx̄ = ________.

"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?"