Chapter 9 Homework



Chapter 9 HomeworkFor each boldface number in Exercises 9.1 to 9.4, (a) state whether it is a parameter or a statistic and (b) use appropriate notation to describe each number, for example QUOTE .1.A carload lot of ball bearings has a mean diameter 2.5003 centimeters (cm). This is within the specifications for acceptance of the lot by the purchaser. By chance, an inspector chooses 100 bearing from the lot that have mean diameter 2.5009 cm. Because this is outside the specified limits, the lot is mistakenly rejected.2.The Bureau of Labor and Statistics last month interviewed 60,000 members of the US labor force, of whom 7.2% were unemployed.3.A telemarketing firm in Los Angeles uses a device that dials residential telephone numbers in that city at random. Of the first 100 numbers dialed, 48% are unlisted. That is not surprising because 52% of all Los Angeles residential phones are unlisted.4.A researcher carried out a randomized comparative experiment with young rats to investigate the effects of a toxic compound in food. She feeds the control group a normal diet. The experimental group receives a diet with 2,500 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.5.Coin tossing can illustrate the idea of a sampling distribution. The population is all outcomes (heads or tails) we would get if we tossed a coin forever. The parameter p is the proportion of heads in this population. We suspect that p is close to 0.5. This is, we think the coin will show about one-half heads in the long run. The sample is the outcomes of 20 tosses, and the statistic QUOTE is the proportion of heads in these 20 tosses.A. Toss a coin 20 times and record the value of QUOTE .B. Repeat the sampling process 10 times. Make a histogram of the 10 values of QUOTE . Is the center of the distribution close to 0.5?C.Ten repetitions give a very crude approximation to the sampling distribution. Repeat the exercise so that you have 40 total trials. Make a histogram of all the values of QUOTE . Is the center close to 0.5? Is the shape approximately normal? 6.Let us illustrate the idea of a sampling distribution of QUOTE in the case of a very small sample from a very small population. The population is the scores of 10 students on an exam:Student0123456789Score82628058727365667462The parameter of interest is the mean score in this population, which is 69.4. The sample is an SRS drawn from the population. Because the students are labeled 0 to 9, a single digit from Table B chooses one student for the sample.A.Use Table B to draw an SRS of size n=4 from this population. Write the four scores in your sample and calculate the mean QUOTE of the sample scores. This statistic is an estimate of the population parameter.B. Repeat this process 10 times. Make a histogram of the 10 values of QUOTE . You are constructing a sampling distribution of QUOTE . Is the center of your histogram close to 69.4?C.Ten repetitions give a very crude approximation to the sampling distribution. Repeat the process until 50 trials are completed. Make a histogram of all the values of QUOTE Is the center close to 69.4? Describe the shape of the distribution. This histogram is a better approximation to the sampling distribution.7.The table below contains the results of simulating on a computer 100 repetitions of the drawing of an SRS of size 200 from a large lot of ball bearings. Ten percent of the bearings in the lot do not conform to the specifications. That is, p=.10 for this population. The numbers in the table are the counts of nonconforming bearings in each sample of 200.1723182715171813161820151816211718191623201818171913272223261713161424221621242130241714161617242116172318232224232320192018202516242424152222162815229191619192524201521252419192028181717251717181918A.Make a table showing how often each occurs. For each count in your table give the corresponding value of the sampling proportion QUOTE = count/200. Then draw a histogram for the values of the statistic QUOTE .B. Describe the shape of the distribution.C.Find the mean of the 100 observations of QUOTE . Mark the mean on your histogram to show its center. Does the statistic QUOTE appear to have large or small bias as an estimate of the population proportion QUOTE p?D. The sampling distribution of QUOTE is the distribution of the values of QUOTE from all possible samples of size 200 from this population. What is the mean of this distribution?E. If we repeatedly selected SRSs of size 1000 instead of 200 from this same population, what would be the mean of the sampling distribution of the same proportion QUOTE ? Would the spread be larger, smaller, or about the same when compared with the spread of your histogram in (A)?8. The Internal Revenue Service plans to examine an SRS of individual federal income tax returns from each state. One variable of interest is the proportion of returns claiming itemized deductions. The total number of tax returns in each state varies from almost 14 million in California to fewer than 210,000 in Wyoming.A.Will the sampling variability of the sample proportion change from state to state if an SRS of 2000 tax returns is selected from each state? Explain your answer.B.Will the sampling variability of the sample proportion change from state to state if an SRS of 1% of all tax returns is selected in each state? Explain your answer.9.An entomologist samples a field for egg masses of a harmful insect by placing a yard-square frame at random locations and examining the ground within the frame carefully. He wants to estimate the proportion of square yards in which egg masses are present. Suppose that in a large field egg masses are present in 20% of all possible yard-square areas. That is, p = 0.2 in this population. A.Use Table B to simulate the presence of absence of egg masses in each square yard of an SRS of 10 square yards from the field. Be sure to explain clearly which digits you used to represent the presence and the absence of egg masses. What proportion of your 10 sample areas had egg masses? This is the statistic QUOTE . B.Repeat (A) with different lines from Table B, until you have simulated the result of 20 SRSs of size 10. What proportion of the square yards in each of your 20 samples had egg masses? Make a stemplot of these 20 values to display the distribution of your 20 observations on QUOTE . What is the mean of the distribution? What is its shape?C.If you looked at all possible SRSs of size 10, rather than just 20 SRSs, what would be the mean of the values of QUOTE ? This is the mean of the sampling distribution of QUOTE .D.In another field, 40% of all square-yard areas contain egg masses. What is the mean of the sampling distribution of QUOTE in the sample from this field?10.A national opinion poll recently estimated that 44% ( QUOTE = .44) of all adults agree that parents of school-age children should be given vouchers good for education at any public or private school of their choice. The polling organization used a probability sampling method for which the sampling proportion QUOTE has a normal distribution with the standard deviation about 0.015. If a sample were drawn by the same method from the state of new Jersey (population 7.8 million) instead of from the entire United States (population 250 million), would this standard deviation be larger, about the same, or smaller? Explain your answer.11.A USA Today poll asked a random sample of 1012 US adults what they do with the milk in the bowl after they have eaten the cereal. Of the respondents, 67% said that they drink it. Suppose that 70% of the US adults actually drink the cereal milk.A.Find the mean and standard deviation of the proportion QUOTE of the sample that say they drink the milk.B. Explain why you can use the formula for the standard deviation of QUOTE in the setting (Rule of Thumb 1)C. Check that you can use the normal approximation for the distribution of QUOTE (Rule of Thumb 2).D. Find the probability of obtaining a sample of 1012 adults in which 67% or fewer say they drink the cereal milk. Do you have any doubts about the results of this poll?E.What sample size would be required to reduce the standard deviation of the sample proportion to one-half the value you found in (A)?F.If the pollsters had surveyed 1012 teenagers instead of 1012 adults, do you think the sample proportion QUOTE would have been greater than, equal to, or less than 0.67? Explain…12.The Gallup Poll asked a probability sample of 1785 adults whether they attended church or synagogue during the past week. Suppose that 40% of the adult population did attend. We would like to know the probability that an SRS of size 1785 would come within plus or minus 3 percentage points of the true value.A.If QUOTE is the proportion of the sample who did attend church or a synagogue, what is the mean of the sampling distribution of QUOTE ? What is the standard deviation?B. Explain why you can use the formula for the standard deviation of QUOTE in this setting (Rule of Thumb 1).C.Check that you can use the normal approximation for the distribution of QUOTE (Rule of Thumb #2).D.Find the probability that QUOTE takes a value between 0.37 and 0.43. Will an SRS of size 1785 usually give a result QUOTE within plus or minus 3 percentage points of the true population proportion? Explain.13.According to a market research firm, 52% of all residential telephone numbers in Los Angeles are unlisted. A telephone sales firm uses random digit dialing equipment that dials residential numbers at random, whether or not they are listed in the telephone directory. The firm calls 500 numbers in Los Angeles. A.What are the mean and standard deviation of the proportion of unlisted numbers in this sample?B.What is the probability that at least half the numbers dialed are unlisted? (Remember to check that you can use the normal approximation.)14.Your mail-order company advertises that it ships 90% of its orders within three working days. You select an SRS of 100 of the 5000 orders received in the past week for an audit. The audit reveals that 86 of these orders were shipped on time. A.What is the sample proportion of orders shipped on time?B.If the company really ships 90% of its orders on time, what is the probability that the proportion in an SRS of 100 orders is as small as the proportion in your sample or smaller?15.Here is a simple probability model for multiple-choice tests. Suppose that a student had probability QUOTE of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher QUOTE than a weaker student.) The correctness of an answer to any specific question doesn’t depend on other questions. A test contains n questions. Then the proportion of correct answers that a student gives is a sample proportion QUOTE from an SRS of size n from a population with population proportion QUOTE . A.Julie is a good student from whom QUOTE . Find the probability that Julie scores 70% or lower on a 100-question test.B. If the test contains 250 questions, what is the probability that Julie will earn 70% or lower?C.How many questions must the test contain in order to reduce the standard deviation of Julie’s proportion of correct answers to half its value of a 100-item test?D.Laura is a weaker student for whom QUOTE Does the answer you gave in (C) for the standard deviation of Julie’s score apply to Laura’s standard deviation also?16.The Helsinki Heart Study asks whether the anticholesterol drug gemfibrozil will reduce heart attacks. In planning such an experiment, the researchers must be confident that the sample sizes are large enough to enable them to observe enough heart attacks. The Helsinki study plans to give gemfibrozil to 2000 men and a placebo to another 2000. The probability of a heart attack during a 5-year period of the study for men his age is about 0.04. We think of the study participants as an SRS from a large population, of which the proportion QUOTE =.04 will have heart attacks.A. Which is the mean number of heart attacks that the study will find in one group of 2000 men if the treatment does not change the probability 0.04?B.What is the probability that the group will suffer at least 75 heart attacks?17.Explain why you cannot use the methods of this section to find the following probabilities.A.A factory employs 3000 unionized workers, of whom 30% are Hispanic. The 15-member union executive committee contains 3 Hispanics. What would be the probability of 3 or fewer Hispanics if the executive committee were chosen at random from all workers?B.A university is concerned about the academic standing of its intercollegiate athletes. A study committee chooses an SRS of 50 of the 316 athletes to interview in detail. Suppose that in fact 40% of the athletes have been told by coaches to neglect their studies on at least one occasion. What is the probability that at least 15 in the sample are among this group?18.The scores of individual students on the American College Testing (ACT) composite college entrance examination have a normal distribution with mean 18.6 and standard deviation 5.9.A.What is the probability that a single student randomly chosen from all those taking test scores 21 or higher?B.Now take an SRS of 50 students who took the test. What are the mean and standard deviation of the average (sample mean) score for the 50 students? Do your results depend on the fact that individual scores have a normal distribution?C.What is the probability that the mean score QUOTE of these students is 21 or higher?19.Juan makes a measurement in a chemistry laboratory and records the result in his lab report. The standard deviation of students’ lab measurements is QUOTE mg.. Juan repeats the measurement 3 times and records the mean QUOTE of his 3 measurements.A. What is the standard deviation of Juan’s mean result? (That is, if Juan kept on making 3 measurements and averaging them, what would be the standard deviation of all his QUOTE ?)B. How many times must Juan repeat the measurement to reduce the standard deviation to QUOTE to 3 milligrams? Explain to someone who knows no statistics the advantage of reporting the average of several measurements rather than a single measurement.20.A study of the health of teenagers plans to measure the blood cholesterol level of an SRS of youth ages 13 to 16 years. The researchers will report the mean QUOTE from their sample as an estimate of the mean cholesterol level QUOTE in this population.A. Explain to someone who knows no statistics what it means to say that QUOTE is an unbiased estimator of QUOTE B. The sample result QUOTE is an unbiased estimator of the population parameter QUOTE no matter what size SRS the study chooses. Explain to someone who knows nothing about statistics why a large sample gives more trustworthy results than a small sample.21. A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a normal distribution with a mean QUOTE = 298 ml and standard deviation QUOTE =3 ml. A. What is the probability that an individual bottle contains less than 295 ml?B.What is the probability that the mean content of the bottles in a six-pack is less than 295 ml? ................
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