AMCS 210 Homework 5 | Complete Solution

AMCS 210 Homework #5

Please start each problem on a new page.
1. An EPA researcher wants to design a study to estimate the mean lead level of sh in a lake located near an industrial area. Based on past sample data, the researcher estimates that for the lead level in the sh population is approximately 0:016 mg/g. He wants to use a 95% con dence interval having a margin of error no greater than 0.005 mg/g.

(a) How many sh does he need to catch?
(b) The researcher collects a sample of n = 50 sh and computes a 95% con dence interval
for the mean lead level to be (0.9932, 1.0045). Interpret this particular interval in the context of the problem.
(c) In your own words, explain the concept of 95% con dence in general for this problem.

2. For a semester project, a student was interested in determining the average weight gain of college students during their freshman year. She took a SRS of 20 freshman and found a sample mean weight gain of x = 5:25 pounds with a standard deviation of 10 pounds. Suppose that it is known that weight gain during the freshman year is normally distributed.
Use this to answer the following questions:

(a) Calculate a 90% CI for , the average weight gain of college students during their
freshman year.
(b) Which of the following is/are correct interpretation(s) for the con dence interval from the previous part? Explain the reasoning for your choice(s).
i. We are 90% con dent that the average weight gain of college students during their
freshman year is in the interval.
ii. We are 90% con dent that 5.25 is in the interval.
iii. There is a 90% probability that 5.25 is in the interval.
iv. Out of 100 intervals, approximately 90% will contain X , and 10 of them will not.

3. Tell in each of the following instances whether the study uses an independent samples or a matched pairs design. In each, identify and de ne the parameter of interest.
(a) Two computing algorithms are compared in terms of the CPU times required to do the same six test problems.
(b) An agronomist compares the yields of two varieties of soybean by planting each variety in 10 separate plots of land (for a total of 20 plots).

(c) An advertising agency has come up with two di erent TV commercials for a household detergent. To determine which one is more e ective, a test is conducted in which a sample of 100 adults is randomly divided into two groups. Each group is shown a di erent commericial, and the people in the group are asked to score the commercial.
(d) Military test pilots who had at least one accident are matched by length of experience to pilots without accidents. The two groups are then surveyed about the number of childhood accidents to determine if the former group of pilots is more \accident prone."

4. To study the e ectiveness of wall insulation in saving energy for home heating, the energy consumption (in MWh) for 10 houses in Bristol, England, was recorded for two winters; the rst winter was before insulation, and the second winter was after insulation. before<-c(12.1, 11.0, 14.1, 13.8, 15.5, 12.2, 12.8, 9.9, 10.8, 12.7) after<-c(12.0, 10.6, 13.4, 11.2, 15.3, 13.6, 12.6, 8.8, 9.6, 12.4)
Assume the energy consumption has normal distribution. Find a 90% con dence interval for the true mean di erence in energy consumption. Does it appear that the wall insulation has reduced the mean energy consumption?

5. In 1993 Time magazine reported a telephone poll survey of 800 adults in the U.S., of whom
45% stated that they had guns in their homes.
(a) Check that the assumptions are met to compute a 95% con dence interval for the true
proportion of adults in the U.S. who have a gun in their home.
(b) Find the 95% con dence interval for the proportion stated in part (a).
(c) Is it guaranteed that the true proportion lies within the bounds that you computed
in part (b)? Explain why or why not.

6. The data in the table below are from a study by chemist and Nobel Laureate Linus Pauling, and it shows the incidence of colds among 279 French skiers who were randomized to the Vitamin C and Placebo Groups.

(a) Compute a 95% CI for the di erence in proportions of skiers who contracted a cold while taking the placebo versus those who took Vitamin C.
(b) Interpret the interval you computed in part (b) in the context of this problem. Cold
Group Yes No Row Total
Vitamin C 17 122 139
Placebo 31 109 140
Column Total 48 231 279