- a)
- Exercise 3.35 (1 Point):
For each of the following sampling situations, identify the population as exactly as possible. That is, say what kind of individuals the population consists of and say exactly which individuals fall in the population. If the information given is not complete, complete the description of the population in a reasonable way.
(a) Each week, the Gallup Poll questions a sample of about 1500 adult U.S. residents in order to discover the opinions of Americans on a wide variety of issues.
(b) Every 10 years, the Census Bureau tries to gather basic information from every household in the United States. But a ``long form'' requesting much additional information is sent to a sample of about 17% of U.S. households.
(c) A machinery manufacturer purchases voltage regulators from a supplier. There are reports that variation in the output voltage of the regulators is affecting the performance of the finished products. To assess the quality of the supplier's production, the manufacturer chooses a sample of 5 regulators from the last shipment for careful laboratory analysis.
- Exercise 3.38 (1 Point):
You want to choose an SRS of 25 of a city's 440 voting precincts for special voting-fraud surveillance on election day. How will you label the 440 precincts? Choose the SRS, and list the precincts you selected. (If you use Table B, enter at line 117 and select only the first 5 precincts in the sample.)
- Exercise 3.41 (1 Point):
Systematic random samples are often used to choose a sample of apartments in a large building or dwelling units in a block at the last stage of a multistage sample. An example will illustrate the idea of a systematic sample. Suppose that we must choose 4 addresses out of 100. Because 100/4 = 25, we can think of the list as four lists of 25 addresses. Choose 1 of the first 25 at random, using Table B. The sample contains this address and the addresses 25, 50, and 75 places down the list from it. If 13 is chosen, for example, then the systematic random sample consists of the addresses numbered 13, 38, 63, and 88.
(a) Use Table B to choose a systematic random sample of 5 addresses from a list of 200. Enter the table at line 120.
(b) Like an SRS, a systematic sample gives all individuals the same chance to be chosen. Explain why this is true, then explain carefully why a systematic sample is nonetheless not an SRS.
- Exercise 3.45 (1 Point):
The list of individuals from which a sample is actually selected is called the sampling frame. Ideally, the frame should list every individual in the population, but in practice this is often difficult. A frame that leaves out part of the population is a common source of undercoverage.
(a) Suppose that a sample of households in a community is selected at random from the telephone directory. What households are omitted from this frame? What types of people do you think are likely to live in these households? These people will probably be underrepresented in the sample.
(b) It is usual in telephone surveys to use random digit dialing equipment that selects the last four digits of a telephone number at random after being given the exchange (the first three digits). Which of the households that you mentioned in your answer to (a) will be included in the sampling frame by random digit dialing?
- Exercise 3.49 (1 Point):
Here are two wordings for the same question. The first question was asked by presidential candidate Ross Perot, and the second by a Time/CNN poll, both in March 1993.A. Should laws be passed to eliminate all possibilities of special interests giving huge sums of money to candidates?
One of these questions drew 40% favoring banning contributions; the other drew 80% with this opinion. Which question produced the 40% and which got 80%? Explain why the results were so different. (W. Mitofsky, ``Mr. Perot, you're no pollster,'' New York Times, March 27, 1993.)
B. Should laws be passed to prohibit interest groups from contributing to campaigns, or do groups have a right to contribute to the candidates they support?
- Exercise 3.50 (1 Point):
Comment on each of the following as a potential sample survey question. Is the question clear? Is it slanted toward a desired response?
(a) ``Does your family use food stamps?''
(b) ``Which of the following best represents your opinion on gun control?1. The government should confiscate our guns.
(c) ``A national system of health insurance should be favored because it would provide health insurance for everyone and would reduce administrative costs.''
2. We have the right to keep and bear arms.''
(d) ``In view of escalating environmental degradation and incipient resource depletion, would you favor economic incentives for recycling of resource-intensive consumer goods?''
- Exercise 3.35 (1 Point):
- b) State whether each boldface number in Exercises 3.51 to 3.54 is a
parameter or a statistic.
- Exercise 3.52 (1 Point):
A carload lot of ball bearings has a mean diameter of 2.503 centimeters (cm). This is within the specifications for acceptance of the lot by the purchaser. The inspector happens to inspect 100 bearings from the lot with a mean diameter of 2.515 cm. This is outside the specified limits, so the lot is mistakenly rejected.
- Exercise 3.54 (1 Point):
A researcher investigating the effects of a toxic compound in food conducts a randomized comparative experiment with young male white rats. A control group is fed a normal diet, while the experimental group is fed a diet with 2500 parts per million of the toxic material. After 8 weeks, the mean weight gain is 335 grams for the control group and 289 grams for the experimental group.
- Exercise 3.55 (1 Point):
Figure 3.10 shows histograms of four sampling distributions of statistics intended to estimate the same parameter. Label each distribution relative to the others as high or low bias and as high or low variability.
- Exercise 3.52 (1 Point):
- c)
- Exercise 4.11 (1 Point):
Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement about an event. (The probability is usually a much more exact measure of likelihood than is the verbal statement.)
0, 0.01, 0.3, 0.6, 0.99, 1
(a) This event is impossible. It can never occur.
(b) This event is certain. It will occur on every trial of the random phenomenon.
(c) This event is very unlikely, but it will occur once in a while in a long sequence of trials.
(d) This event will occur more often than not.
- Exercise 4.12 (1 Point):
In each of the following situations, describe a sample space S for the random phenomenon. In some cases, you have some freedom in your choice of S.
(a) A seed is planted in the ground. It either germinates or fails to grow.
(b) A patient with a usually fatal form of cancer is given a new treatment. The response variable is the length of time that the patient lives after treatment.
(c) A student enrolls in a statistics course and at the end of the semester receives a letter grade.
(d) A basketball player shoots two free throws.
(e) A year after knee surgery, a patient is asked to rate the amount of pain in the knee. A seven-point scale is used, with 1 corresponding to no pain and 7 corresponding to extreme discomfort.
- Exercise 4.15 (1 Point):
All human blood can be typed as one of O, A, B, or AB, but the distribution of the types varies a bit with race. Here is the distribution of the blood type of a randomly chosen black American:(a) What is the probability of type AB blood? Why?Blood type O A B AB Probability 0.49 0.27 0.20 ?
(b) Maria has type B blood. She can safely receive blood transfusions from people with blood types O and B. What is the probability that a randomly chosen black American can donate blood to Maria?
- Exercise 4.17 (1 Point):
Figure 4.5 displays several assignments of probabilities to the six faces of a die. We can learn which assignment is actually accurate for a particular die only by rolling the die many times. However, some of the assignments are not legitimate assignments of probability. That is, they do not obey the rules. Which are legitimate and which are not? In the case of the illegitimate models, explain what is wrong.
- Exercise 4.22 (1 Point):
Choose an acre of land in Canada at random. The probability is 0.35 that it is forest and 0.03 that it is pasture.
(a) What is the probability that the acre chosen is not forested?
(b) What is the probability that it is either forest or pasture?
(c) What is the probability that a randomly chosen acre in Canada is something other than forest or pasture?
- Exercise 4.24 (1 Point):
Choose an American farm at random and measure its size in acres. Here are the probabilities that the farm chosen falls in several acreage categories:Let A be the event that the farm is less than 50 acres in size, and let B be the event that it is 500 acres or more.Acres < 10 10-49 50-99 100-179 180-499 500-999 1000-1999 >= 2000 Probability 0.09 0.20 0.15 0.16 0.22 0.09 0.05 0.04
(a) Find P(A) and P(B).
(b) Describe Ac in words and find P(Ac) by the complement rule.
(c) Describe {A or B} in words and find its probability by the addition rule.
- Exercise 4.26 (1 Point):
A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors.
(a) What is the probability that the ball will land in any one slot?
(b) If you bet on ``red,'' you win if the ball lands in a red slot. What is the probability of winning?
(c) The slot numbers are laid out on a board on which gamblers place their bets. One column of numbers on the board contains all multiples of 3, that is, 3, 6, 9, ..., 36. You place a ``column bet'' that wins if any of these numbers comes up. What is your probability of winning?
- Exercise 4.30 (1 Point):
Government data show that 27% of the civilian labor force have at least 4 years of college and that 16% of the labor force work as laborers or operators of machines or vehicles. Can you conclude that because (0.27)(0.16) = 0.043, about 4% of the labor force are college-educated laborers or operators? Explain your answer.
- Exercise 4.31 (1 Point):
Choose at random a U.S. resident at least 25 years of age. We are interested in the eventsA = {The person chosen completed 4 years of college}
Government data recorded in Table 2.14 on page 194 (from Edition 3 - see handout!) allow us to assign probabilities to these events.
B = {The person chosen is between 55 years old or older}
(a) Explain why P(A) = 0.230.
(b) Find P(B).
(c) Find the probability that the person chosen is at least 55 years old and has 4 years of college education, P(A and B). Are the events A and B independent?
- Exercise 4.36 (1 Point):
The type of medical care a patient receives may vary with the age of the patient. A large study of women who had a breast lump investigated whether or not each woman received a mammogram and a biopsy when the lump was discovered. Here are some probabilities estimated by the study. The entries in the table are the probabilities that both of two events occur; for example, 0.321 is the probability that a patient is under 65 years of age and the tests were done. The four probabilities in the table have sum 1 because the table lists all possible outcomes.(a) What is the probability that a patient in this study is under 65? That a patient is 65 or over?Tests done? Yes No Age under 65 0.321 0.124 Age 65 or over 0.365 0.190
(b) What is the probability that the tests were done for a patient? That they were not done?
(c) Are the events A = {the patient was 65 or older} and B = {the tests were done} independent? Were the tests omitted on older patients more or less frequently than would be the case if testing were independent of age?
- Exercise 4.11 (1 Point):
- d) These last 4 exercises from Section 2 are review
questions that should
be part of your preparation for Midterm 2.
When you work on these questions, be as
efficient as possible. When you have to calculate the correlation or
the least squares regression line, first calculate the sums of the
x's, x2's, y's, y2's, and xy's,
then determine SS(x), SS(y), and
SS(xy), and finally determine the regression coefficients and
the correlation coefficient r. If you cannot recall this fast solution,
check the lecture notes. Also, look
at the solutions for Homework Assignment 5.
Alternatively, learn how to use your calculator efficiently.
- Exercise 2.19 (2 Points):
The table below (from Exercise 2.10) gives data on the lean body mass and metabolic rate for 12 women and 7 men:Subject Sex Mass Rate Subject Sex Mass Rate 1 M 62.0 1792 11 F 40.3 1189 2 M 62.9 1666 12 F 33.1 913 3 F 36.1 995 13 M 51.9 1460 4 F 54.6 1425 14 F 42.4 1124 5 F 48.5 1396 15 F 34.5 1052 6 F 42.0 1418 16 F 51.1 1347 7 M 47.4 1362 17 F 41.2 1204 8 F 50.6 1502 18 M 51.9 1867 9 F 42.0 1256 19 M 46.9 1439 10 M 48.7 1614
(a) If you did not do Exercise 2.10, make a scatterplot. Use different symbols or colors for women and men. Do you think the correlation will be about the same for men and women or quite different for the two groups? Why?
(b) Find r for women alone and also for men alone.
(c) Calculate the mean body mass for the women and for the men. Does the fact that the men are heavier than the women on the average influence the correlations? If so, in what way?
(d) Lean body mass was measured in kilograms. How would the correlations change if we measured body mass in pounds? (There are about 2.2 pounds in a kilogram.)
- Exercise 2.32 (1 Point):
Each of the following statements contains a blunder. Explain in each case what is wrong.
(a) ``There is a high correlation between the gender of American workers and their income.''
(b) ``We found a high correlation (r = 1.09) between students' ratings of faculty teaching and ratings made by other faculty members.''
(c) ``The correlation between planting rate and yield of corn was found to be r = 0.23 bushel.''
- Exercise 2.35 (1 Point):
(Review of straight lines) During the period after birth, a male white rat gains 40 grams (g) per week. (This rat is unusually regular in his growth, but 40 g per week is a realistic rate.)
(a) If the rat weighed 100 g at birth, give an equation for his weight after weeks. What is the slope of this line?
(b) Draw a graph of this line between birth and 10 weeks of age.
(c) Would you be willing to use this line to predict the rat's weight at age 2 years? Do the prediction and think about the reasonableness of the result. (There are 454 g in a pound. To help you assess the result, note that a large cat weighs about 10 pounds.)
- Exercise 2.111 (2 Points):
Lamb's-quarter is a common weed that interferes with the growth of corn. An agriculture researcher planted corn at the same rate in 16 small plots of ground, then weeded the plots by hand to allow a fixed number of lamb's-quarter plants to grow in each meter of corn row. No other weeds were allowed to grow. Here are the yields of corn (bushels per acre) in each of the plots (data provided by Samuel Phillips, Purdue University):(a) What are the explanatory and response variables in this experiment?Weeds Corn Weeds Corn Weeds Corn Weeds Corn per meter yield per meter yield per meter yield per meter yield 0 166.7 1 166.2 3 158.6 9 162.8 0 172.2 1 157.3 3 176.4 9 142.4 0 165.0 1 166.7 3 153.1 9 162.8 0 176.9 1 161.1 3 156.0 9 162.4
(b) Make side-by-side stemplots of the yields, after rounding to the nearest bushel. What do you conclude about the effect of this weed on corn yield?
(c) Make a scatterplot of corn yield against weeds per meter. Find the least-squares regression line and add it to your plot. The advantage of regression over the side-by-side comparison in (b) is that we can use the fitted model to draw conclusions for counts of weeds other than the ones the researcher actually used. What does the slope of the fitted line tell us about the effect of lamb's-quarter on corn yield?
(d) Predict the yield for corn grown under these conditions with 6 lamb's-quarter plants per meter of row.
- Exercise 2.19 (2 Points):