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introduction to probability statistics
Introduction To Probability And Statistics 3rd Edition William Mendenhall - Solutions
8.74 Suppose you wish to estimate a population mean based on a random sample of n observations, and prior experience suggests that or 12.7. If you wish to estimate correct to within 1.6, with probability equal to 0.95, how many observations should be included in your sample?
8.75 Suppose you wish to estimate a binomial parameter p correct to within 0.04, with probability equal to 0.95. If you suspect that p is equal to some value between 0.1 and 0.3 and you want to be certain that your sample is large enough, how large should n be? (HINT: When calculating the standard
8.76 Independent random samples of n = n = n observations are to be selected from each of two populations 1 and 2. If you wish to estimate the difference between the two population means correct to within 0.17, with probability equal to 0.90, how large should n and n be? Assume that you know ==27.8.
8.77 Independent random samples of n = n = n observations are to be selected from each of two bino- mial populations 1 and 2. If you wish to estimate the difference in the two population proportions correct to within 0.05, with probability equal to 0.98, how large should n be? Assume that you have
8.78 Operating Expenses A random sampling of a company's monthly operating expenses for n = 36 months produced a sample mean of $5474 and a standard deviation of $764. Find a 90% upper confidence bound for the company's mean monthly expenses.
8.79 Fear Factor II Exercise 8.21 discussed a research poll done by EKOS Research to determine whether Canadians think the world is more dangerous today than 25 years ago. Suppose you were designing a poll of this type.a. Explain how you would select your sample. What problems might you encounter
8.80 Political Corruption A questionnaire is designed to investigate attitudes about political corruption in government. The experimenter would like to survey two different groups-Liberals and Conservatives and compare the responses to various "yes- no" questions for the two groups. The
8.81 Less Red Meat! Canadians are becoming more conscious of the importance of good nutrition, and some researchers believe that we may be altering our diets to include less red meat and more fruits and vegetables. To test this theory, a researcher decides to select hospital nutritional records for
8.82 Red Meat, continued Refer to Exercise 8.81. The researcher selects two groups of 400 subjects each and collects the following sample information on the annual beef consumption now and 10 years ago:Â a. The researcher would like to show that per-capita beef consumption has decreased in the last
8.83 Hunting Season If a wildlife service wishes to estimate the mean number of days of hunting per hunter for all hunters licensed in the province during a given season, with a bound on the error of estimation equal to 2 hunting days, how many hunters must be included in the survey? Assume that
8.84 Polluted Rain Suppose you wish to estimate the mean pH of rainfalls in an area that suffers heavy pollution due to the discharge of smoke from a power plant. You know that or is in the neighbourhood of 0.5 pH, and you wish your estimate to lie within 0.1 of , with a probability near 0.95.
8.85 pH in Rainfall Refer to Exercise 8.84. Suppose you wish to estimate the difference between the mean acidity for rainfalls at two different locations, one in a relatively unpolluted area along the ocean and the other in an area subject to heavy air pollution. If you wish your estimate to be
8.86 GPAs You want to estimate the difference in grade point averages between two groups of univer- sity students accurate to within 0.2 grade point, with probability approximately equal to 0.95. If the standard deviation of the grade point measurements is approxi- mately equal to 0.6, how many
8.87 Selenium, again Refer to the comparison of the daily adult intake of selenium in two different regions of Canada in Exercise 8.48. Suppose you wish to estimate the difference in the mean daily intakes between the two regions correct to within 5 micro- grams, with probability equal to 0.90. If
8.88 State the Central Limit Theorem. Of what value is the Central Limit Theorem in large-sample statistical estimation?
8.90 Independent random samples of n = 50 and n = 60 observations were selected from populations 1 and 2, respectively. The sample sizes and computed sample statistics are given in the table:Â Find a 90% confidence interval for the difference in population means and interpret the interval. Sample
8.91 Refer to Exercise 8.90. Suppose you wish to estimate (u) correct to within 0.2, with probabil- ity equal to 0.95. If you plan to use equal sample sizes, how large should n, and n be?
8.93 Refer to Exercise 8.92. How large a sample is required if you wish to estimate p correct to within 0.025, with probability equal to 0.90?
8.94 Independent random samples of n = 40 and n=80 observations were selected from binomial popu- lations 1 and 2, respectively. The number of successes in the two samples were x = 17 and x2 = 23. Find a 99% confidence interval for the difference between the two binomial population proportions.
8.95 Refer to Exercise 8.94. Suppose you wish to estimate (p1 - p) correct to within 0.06, with probability equal to 0.99, and you plan to use equal sample sizes that is, n = n2. How large should n and n be?
8.96 Ethnic Cuisine Ethnic groups in North America buy differing amounts of various food prod- ucts because of their ethnic cuisine. Asians buy fewer canned vegetables than do other groups, and Hispanics purchase more cooking oil. A researcher interested in market segmentation for these two groups
8.97 Women on Wall Street Women on Wall Street can earn large salaries, but may need to make sacrifices in their personal lives. In fact, many women in the securities industry have to make significant personal sacrifices. A survey of 482 women and 356 men found that only half of the women have
8.98 Smoking and Blood Pressure An experiment was conducted to estimate the effect of smoking on the blood pressure of a group of 35 cigarette smokers. The difference for each participant was obtained by taking the difference in the blood pressure readings at the beginning of the experiment and
8.99 Blood Pressure, continued Using a confi- dence coefficient equal to 0.90, place a confidence interval on the mean increase in blood pressure for Exercise 8.98.
8.100 lodine Concentration Based on repeated measurements of the iodine concentration in a solution, a chemist reports the concentration as 4.614 moles/litre, with an "error margin of 0.006."a. How would you interpret the chemist's "error margin"?b. If the reported concentration is based on a
8.101 Heights If it is assumed that the heights of men are normally distributed, with a standard devia- tion of 6 cm, how large a sample should be taken to be fairly sure (probability 0.95) that the sample mean does not differ from the true mean (population mean) by more than 0.50 in absolute value?
8.102 Chicken Feed An experimenter fed different rations, A and B, to two groups of 100 chicks each. Assume that all factors other than rations are the same for both groups. Of the chicks fed ration A, 13 died, and of the chicks fed ration B, 6 died.a. Construct a 98% confidence interval for the
8.103 Antibiotics You want to estimate the mean hourly yield for a process that manufactures an antibi- otic. You observe the process for 100 hourly periods chosen at random, with the results = 34 ounces per hour and s = 3. Estimate the mean hourly yield for the process using a 95% confidence
8.104 Cheese and Beer The average European has become accustomed to eating away from home, especially at fast-food restaurants. Partly as a result of this fast-food habit, the per-capita consumption of cheese (the main ingredient in pizza) and beer has risen dramatically from a decade ago. A study
8.105 Healthy Eating Don't Americans know that eating pizza and french fries leads to being overweight? In an American Demographics article23, a survey of women who are the main meal preparers in their house- holds reported these results: 90% know that obesity causes health problems. 80% know that
8.106 Sunflowers In an article in the Annals of Bot- any, a researcher reported the basal stem diameters of two groups of dicot sunflowers: those that were left to sway freely in the wind and those that were artificially supported.24 A similar experiment was conducted for monocot maize plants.
8.107 Working in Retirement Research released by Investors Group shows that non-retired Canadians overwhelmingly say their physical health is better than their financial health (67%) and many may be relying on continued good health to enable them to keep working during their retirement years.25 The
8.108 University Costs A university administra- tor wishes to estimate the average cost of the first year at a particular university correct to within $500, with a probability of 0.95. If a random sample of first-year students is to be selected and each asked to keep financial data, how many must
8.109 Quality Control A quality-control engineer wants to estimate the fraction of defectives in a large lot of film cartridges. From previous experience, he feels that the actual fraction of defectives should be somewhere around 0.05. How large a sample should he take if he wants to estimate the
8.110 Circuit Boards Samples of 400 printed circuit boards were selected from each of two production lines A and B. Line A produced 40 defectives, and line B produced 80 defectives. Estimate the difference in the actual fractions of defectives for the two lines with a confidence coefficient of 0.90.
8.111 Circuit Boards II Refer to Exercise 8.110. Suppose 10 samples of n = 400 printed circuit boards were tested and a confidence interval was constructed for p for each of the ten samples. What is the prob- ability that exactly one of the intervals will not contain the true value of p? That at
8.112 Ice Hockey The ability to accelerate rapidly is an important attribute for an ice hockey player. G. Wayne Marino investigated some of the variables related to the acceleration and speed of a hockey player from a stopped position.26 Sixty-nine hockey players, varsity and intramural, were
8.113 Ice Hockey II Exercise 8.112 presented sta- tistics from a study of fast starts by ice hockey skaters. The mean and standard deviation of the 69 individual average acceleration measurements over the 6 m dis- tance were 2.962 and 0.529 metres per second (m/sec), respectively.a. Find a 95%
8.114 Ice Hockey III The mean and standard devia- tion of the speeds of the sample of 69 skaters at the end of the 6 m distance in Exercise 8.112 were 5.753 and 0.892 (m/sec) respectively.a. Find a 95% confidence interval for the mean veloc- ity at the 6 m mark. Interpret the interval.b. Suppose
8.115 Student Loan Debt In a study done, the inci- dence of university student loan debt in Alberta was found to be 48.6% as compared to the national average of 42%.27 The amount of student debt among Alberta graduates was $17,900, less than the national average of $18,900. A group of n = 49
8.116 Recidivism An experimental rehabilitation technique was used on released convicts. It was shown that 79 of 121 men subjected to the technique pursued useful and crime-free lives for a three-year period fol- lowing prison release. Find a 95% confidence interval for p, the probability that a
8.117 Specific Gravity If 36 measurements of the specific gravity of aluminum had a mean of 2.705 and a standard deviation of 0.028, construct a 98% confidence interval for the actual specific gravity of aluminum.
8.118 Audiology Research In a study to establish the absolute threshold of hearing, 70 male first-year university students were asked to participate. Each sub- ject was seated in a soundproof room and a 150 Hz tone was presented at a large number of stimulus levels in a randomized order. The
8.119 Right-or Left-Handed A researcher classi- fied his subjects as innately right-handed or left-handed by comparing thumbnail widths. He took a sample of 400 men and found that 80 men could be classified as left-handed according to his criterion. Estimate the pro- portion of all males in the
8.120 The Citrus Red Mite An entomologist wishes to estimate the average development time of the citrus red mite correct to within 0.5 day. From previous experiments it is known that or is in the neighbourhood of 4 days. How large a sample should the entomologist take to be 95% confident of her
8.121 The Citrus Red Mite, continued A grower believes that one in five of his citrus trees are infected with the citrus red mite, mentioned in Exercise 8.120. How large a sample should be taken if the grower wishes to estimate the proportion of his trees that are infected with citrus red mite to
1. Verify the margin of error of 2.2 percentage points given by the survey designers for the full sample of 2000 adults. Find the margin of error for the two split samples of 1000 adults. What elements, if any, go into a Canadian "national identity" has long been a ques- tion of interest to
2. Do the numbers reported in the table represent the number of people who fell into those categories? If not, what do those numbers represent?What elements, if any, go into a Canadian "national identity" has long been a ques- tion of interest to scholars and ordinary citizens alike. The Centre for
3. When the set of questions on pride in Canada were asked, the pollster rotated the order of things and events given to the respondent. Why do you suppose this tech- nique was used?What elements, if any, go into a Canadian "national identity" has long been a ques- tion of interest to scholars and
4. Construct 95% confidence intervals for the proportion of Canadians who:a. report that they were "proud" when Canada decided to not participate in the war on Iraq.b. report that they were "not proud" about this decision.What elements, if any, go into a Canadian "national identity" has long been a
5. Compare the percentage of people in the full sample of 2000 who say that mul- ticulturalism makes them proud to be Canadian with the percentage of those in the second split sample who say that "the fact that people from different cultural groups in Canada get along and live in peace" makes them
6. If these questions were asked today, would you expect the responses to be similar to those reported here or would you expect them to differ substantially?What elements, if any, go into a Canadian "national identity" has long been a ques- tion of interest to scholars and ordinary citizens alike.
North York General Hospital, a multi-site community teaching hospital, gets approxi- mately 75,000 visits to its emergency department each year. With that many patients, Emergency was often very overcrowded and patients weren't flowing through care as they should. North York General Hospital used
A study reveals that interest among Canadian women in home and birthing centre delivery and in midwife care for women at low risk has been growing recently. In the past few years in Canada, several provincial governments have legislated midwife care, and other provinces are considering doing so. A
7.48 A random sample of size n = 50 is selected from a binomial distribution with population proportion p = 0.7.a. What will be the approximate shape of the sampling distribution of p?b. What will be the mean and standard deviation (or standard error) of the sampling distribution of p?c. Find the
7.62 Black Jack A gambling casino records and plots the mean daily gain or loss from five blackjack tables on an x chart. The overall mean of the sample means and the standard deviation of the combined data over 40 weeks were $10,752 and s = $1605, respectively.a. Construct an x chart for the mean
7.63 Brass Rivets A producer of brass rivets randomly samples 400 rivets each hour and calculates the proportion of defectives in the sample. The mean sample proportion calculated from 200 samples was equal to 0.021. Construct a control chart for the propor- tion of defectives in samples of 400
7.64 Lumber Specs The manager of a building-supplies company randomly samples incoming lumber to see whether it meets quality specifications. From each shipment, 100 pieces of 2 x 4 lumber are inspected and judged according to whether they are first (acceptable) or second (defec- tive) grade. The
7.65 Coal Burning Power Plant A coal-burning power plant tests and measures three specimens of coal each day to monitor the percentage of ash in the coal. The overall mean of 30 daily sample means and the combined standard deviation of all the data were =7.24 and s = 0.07, respectively. Construct
7.66 Nuclear Power Plant The data in the EX0766 table are measures of the radiation in air particu- lates at a nuclear power plant. Four measurements were recorded at weekly intervals over a 26-week period. Use the data to construct an x chart and plot the 26 values of x. Explain how the chart can
7.67 Baseball Bats A hardwoods manufacturing plant has several different production lines to make baseball bats of different weights. One such produc- tion line is designed to produce bats weighing 910 grams. During a period of time when the production process was known to be in statistical
7.68 Baseball Bats, continued Refer to Exercise 7.67 and suppose that during a day when the state of the 910-gram bat production process was unknown, the following measurements were obtained at hourly intervals.Each measurement represents a statistic computed from a sample of five bat weights
7.70 Refer to Exercise 7.69. Find the sampling distribution for x if random samples of size n = 3 are selected without replacement. Graph the sampling distribution of x.
7.71 Suppose a random sample of n = 5 observations is selected from a population that is normally distrib- uted, with mean equal to 1 and standard deviation equal to 0.36.a. Give the mean and standard deviation of the sam- pling distribution of .x.b. Find the probability that exceeds 1.3.c. Find
7.72 Batteries A certain type of automobile bat- tery is known to last an average of 1110 days with a standard deviation of 80 days. If 400 of these batteries are selected, find the following probabilities for the average length of life of the selected batteries:a. The average is between 1100 and
7.59 Explain the difference between an x chart and a p chart.
7.49 A random sample of size n = 80 is selected from a binomial distribution with population proportion p = 0.25.a. What will be the approximate shape of the sampling distribution of p?b. What will be the mean and standard deviation (or standard error) of the sampling distribution of p?c. Find the
7.50a. Is the normal approximation to the sampling distribution of p appropriate when n = 400 and p = 0.8?b. Use the results of part a to find the probability that p is greater than 0.83.c. Use the results of part a to find the probability that plies between 0.76 and 0.84.
7.51 Road Trip! Parents with children list a GPS system (28%) and a DVD player (28%) as "must have" accessories for a road trip. 12 Suppose a sample of n = 1000 parents are randomly selected and asked what devices they would like to have for a family road trip. Let p be the proportion of parents in
7.52 Fats and Sweets According to a study in the American Journal of Public Health, diets high in fat, sugar, and grains were associated with lower diet costs after adjustment for energy intakes, gender, and age. For most levels of energy intake, each additional 100 g of fat and sweets was
7.53 Surfing the Net Do you use the Internet to gather information for a project? A survey reports that the per- centage of students who used the Internet as their major resource for a school project in a recent year was 66%.14 Suppose that you take a sample of n = 1000 students, and record the
7.54 M&Ms According to the M&M website, the average percentage of brown M&M candies in a pack- age of milk chocolate M&Ms is 13%.15 (This percentage varies, however, among the different types of packaged M&Ms.) Suppose you randomly select a package of milk chocolate M&Ms that contains 55 candies
7.55 Measured Obesity Over the past several years, the prevalence of overweight and obese children and adolescents has risen, with the most substantial increases observed in economically developed coun- tries. 16 According to the results of the 2004 Canadian Community Health Survey: Nutrition
7.56 Oh, Nuts! Are you a chocolate "purist," or do you like other ingredients in your chocolate? American Demographics reports that almost 75% of consum- ers like traditional ingredients such as nuts or caramel in their chocolate. They are less enthusiastic about the taste of mint or coffee that
A statistical process control monitoring system samples the inside diameters of n = 4 bearings each hour. Table 7.6 provides the data for k = 25 hourly samples. Construct an x chart for monitoring the process mean.
A manufacturer of ballpoint pens randomly samples 400 pens per day and tests each to see whether the ink flow is acceptable. The proportions of pens judged defective each day over a 40-day period are listed in Table 7.7. Construct a control chart for the proportion p defective in samples of n = 400
7.73 Lead Pipes Studies indicate that drinking water supplied by some old lead-lined city piping sys- tems may contain harmful levels of lead. An important study of the Boston water supply system showed that the distribution of lead content readings for individual water specimens had a mean and
7.74 Biomass The total amount of vegetation held by the earth's forests is important to both ecologists and politicians because green plants absorb carbon dioxide. An underestimate of the earth's vegetative mass, or biomass, means that much of the carbon dioxide emitted by human activities
7.89 Canned Tomatoes During long pro- duction runs of canned tomatoes, the average weights (in mL) of samples of five cans of standard- grade tomatoes in puree form were taken at 30 control points during an 11-day period. These results are shown in the table. 24 When the machine is performing
7.90 Pepsi or Coke? The battle for consumer preference continues between Pepsi and Coke. How can you make your preferences known? There is a webpage where you can vote for one of these colas if you click on the link that says PAY CASH for your opinion. Explain why the respondents do not repre- sent
7.91 Strawberries An experimenter wants to find an appropriate temperature at which to store fresh strawberries to minimize the loss of ascorbic acid. There are 20 storage containers, each with controllable temperature, in which strawberries can be stored. If two storage temperatures are to be
7.92 Filling Pop Cans A bottler of soft drinks pack- ages cans in six-packs. Suppose that the fill per can has an approximate normal distribution with a mean of 355 mL and a standard deviation of 5.91 mL.a. What is the distribution of the total fill for a case of 24 cans?b. What is the probability
7.93 Total Packing Weight Packages of food whose average weight is 454 grams with a standard deviation of 17 grams are shipped in boxes of 24 packages. If the package weights are approximately normally distrib- uted, what is the probability that a box of 24 packages will weigh more than 11,113
7.94 Electronic Components A manufacturing process is designed to produce an electronic compo- nent for use in small portable television sets. The components are all of standard size and need not conform to any measurable characteristic, but are sometimes inoperable when emerging from the
1. To evaluate the results of Seligman's Monte Carlo experiment, first find the prob- ability distribution of the gain x on a single $5 bet.The technique of simulating a process that contains random elements and repeating the process over and over to see how it behaves is called a Monte Carlo
2. Find the expected value and variance of the gain x from part 1. The technique of simulating a process that contains random elements and repeating the process over and over to see how it behaves is called a Monte Carlo procedure. It is widely used in business and other fields to investigate the
3. Find the expected value and variance for the evening's gain, the sum of the gains or losses for the 200 bets of $5 each. The technique of simulating a process that contains random elements and repeating the process over and over to see how it behaves is called a Monte Carlo procedure. It is
4. Use the results of part 2 to evaluate the probability of 7 out of 365 evenings result- ing in a loss of the total $1000 stake. The technique of simulating a process that contains random elements and repeating the process over and over to see how it behaves is called a Monte Carlo procedure. It
5. Use the results of part 3 to evaluate the probability that the largest evening's win- nings were as great as $1160. The technique of simulating a process that contains random elements and repeating the process over and over to see how it behaves is called a Monte Carlo procedure. It is widely
Canada's profile in the international physics community got a huge boost as renowned "superstar" cosmologist Stephen Hawking (shown in the photo below floating on a zero-gravity jet) accepted a research post at the country's "crown jewel" of theoretical physics study, The Perimeter Institute in
7.88 Light bulbs, again Refer to Exercise 7.86. During a given week the number of defective bulbs in each of five samples of 100 were found to be 2, 4, 9, 7, and 11. Is there reason to believe that the production process has been producing an excessive proportion of defectives at any time during
7.87 Light bulbs, continued A hardware store chain purchases large shipments of light bulbs from the manufacturer described in Exercise 7.86 and speci- fies that each shipment must contain no more than 4% defectives. When the manufacturing process is in con- trol, what is the probability that the
7.75 Hard Hats The safety requirements for hard hats worn by construction workers and others, estab- lished by the American National Standards Institute (ANSI), specify that each of three hats pass the following test. A hat is mounted on an aluminum head form. An 4-kilogram (kg) steel ball is
7.76 Imagery and Memory A research psycholo- gist is planning an experiment to determine whether the use of imagery picturing a word in your mind- affects people's ability to memorize. He wants to use two groups of subjects: a group that memorizes a set of 20 words using the imagery technique, and
7.77 Child Abuse A study of nearly 2000 women included questions dealing with child abuse and its effect on the women's adult life. The study reported on the likelihood that a woman who was abused as a child would suffer either physical abuse or physical problems arising from depression, anxiety,
7.78 Sprouting Radishes A biology experiment was designed to determine whether sprouting radish seeds inhibit the germination of lettuce seeds. Three 10-centimetre petri dishes were used. The first con- tained 26 lettuce seeds, the second contained 26 radish seeds, and the third contained 13
7.79 Canadian Identity Do Canadians see them- selves as Canadians, as a province, or equally as a Canadian and a province? According to Strategic Counsel who presented the findings of the Globe and Mail/CTV 2005 polling program, 41% said they see themselves as Canadians. 23 The result is based on
7.80 Telephone Service Suppose a telephone com- pany executive wishes to select a random sample of n = 20 (a small number is used to simplify the exer- cise) out of 7000 customers for a survey of customer attitudes concerning service. If the customers are num- bered for identification purposes,
7.81 Rh-Positive The proportion of individuals with an Rh-positive blood type is 85%. You have a random sample of n = 500 individuals.a. What are the mean and standard deviation of p, the sample proportion with Rh-positive blood type?b. Is the distribution of approximately normal? Jus- tify your
7.82 What survey design is used in each of these situations?a. A random sample of n = 50 city blocks is selected, and a census is done for each single-family dwelling on each block.b. The highway patrol stops every tenth vehicle on a given city artery between 9:00 A.M. and 3:00 P.M. to perform a
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