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introduction to probability statistics
Introduction To Probability And Statistics 3rd Edition William Mendenhall - Solutions
10.105 Chemical Purity A chemical manufacturer claims that the purity of its product never varies by more than 2%. Five batches were tested and given purity readings of 98.2, 97.1, 98.9, 97.7, and 97.9%.a. Do the data provide sufficient evidence to contradict the manufacturer's claim? (HINT: To be
10.106 454-gram Cans? A cannery prints "weight 454 grams" on its label. The quality control supervisor selects nine cans at random and weighs them. He finds 445 and s 14.2. Do the data present sufficient evidence to indicate that the mean weight is less than that claimed on the label?
10.107 Reaction Time III A psychologist wishes to verify that a certain drug increases the reaction time to a given stimulus. The following reaction times (in tenths of a second) were recorded before and after injection of the drug for each of four subjects:Test at the 5% level of significance to
10.108 Food Production At a time when energy conservation is so important, some scientists think closer scrutiny should be given to the cost (in energy) of producing various forms of food. Suppose you wish to compare the mean amount of oil required to produce 4047 square metres of corn versus 4047
10.109 Alcohol and Altitude The effect of alcohol consumption on the body appears to be much greater at high altitudes than at sea level. To test this theory, a scientist randomly selects 12 subjects and randomly divides them into two groups of six each. One group is put into a chamber that
10.110 Stock Risks The closing prices of two common stocks were recorded for a period of 15 days. The means and variances area. Do these data present sufficient evidence to indi- cate a difference between the variabilities of the closing prices of the two stocks for the populations associated with
10.111 Auto Design An experiment is conducted to compare two new automobile designs. Twenty people are randomly selected, and each person is asked to rate each design on a scale of 1 (poor) to 10 (excellent). The resulting ratings will be used to test the null hypothesis that the mean level of
10.112 Safety Programs The data shown here were collected on lost-time accidents (the figures given are mean work-hours lost per month over a period of one year) before and after an industrial safety program was put into effect. Data were recorded for six indus- trial plants. Do the data provide
10.113 Two Different Entrees To compare the demand for two different entrees, the manager of a cafeteria recorded the number of purchases of each entree on seven consecutive days. The data are shown in the table. Do the data provide sufficient evidence to indicate a greater mean demand for one of
10.114 Pollution Control The U.S. Environmental Protection Agency limit on the allowable discharge of suspended solids into rivers and streams is 60 milligrams per litre (mg/L) per day. A study of water samples selected from the discharge at a phosphate mine shows that over a long period, the mean
10.115 Enzymes Two methods were used to measure the specific activity (in units of enzyme activity per milligram of protein) of an enzyme. One unit of enzyme activity is the amount that catalyzes the formation of one micromole of product per minute under specified conditions. Use an appropriate
10.116 Connector Rods A producer of machine parts claimed that the diameters of the connector rods produced by its plant had a variance of at most 0.173 cm. A random sample of 15 connector rods from the plant produced a sample mean and variance of 1.4 cm and 0.58 cm, respectively.a. Is there
10.117 Sleep and the University Student How much sleep do you get on a typical study night? A group of 10 university students were asked to report the number of hours that they slept on the previous night with the following results:a. Find a 99% confidence interval for the average number of hours
10.118 Arranging Objects The following data are the response times in seconds for n = 25 first graders to arrange three objects by size.Find a 95% confidence interval for the average re- sponse time for first graders to arrange three objects by size. Interpret this interval. 5.2 3.8 5.7 3.9 3.7 4.2
10.119 The NBA Finals Want to attend a pro-basketball finals game? The average prices for the NBA rematch of the Boston Celtics and the L.A. Lakers in 2010 compared to the average ticket prices in 2008 are given in the following table.24a. If we were to assume that the prices given in the table
10.120 Finger-Lickin' Good! Maybe too good, according to tests performed by the consumer testing division of Good Housekeeping. Nutritional informa- tion provided by KFC claims that each small bag of potato wedges contains 136 g of food, for a total of 280 calories. A sample of 10 orders from KFC
10.121 Mall Rats One study investigated consumer habits at the mall. We tend to spend the most money shopping on the weekends, and, in particular, on Sundays from 4 to 6 P.M. Wednesday morning shop- pers spend the least!26 Suppose that a random sample of 20 weekend shoppers and a random sample of
10.122 Border Wars As the costs of prescription drugs escalate, more and more American senior citizens are ordering prescriptions from Canada, or actually crossing the border to buy prescription drugs. The price of a typical prescription for nine best-selling drugs was recorded at randomly selected
1. American martens feed on a variety of small mammals, birds, insects, fish, carrion, and vegetation. Marten food habits differ substantially in different geographic areas. Do the data indicate that the jam-lard-fish oil mixture attracts the martens in the Temagami region more efficiently than
2. Peanut butter is one of the favourite foods of North Americans. Do American martens have a taste for peanut butter as well? Answer by comparing the effect of peanut butter and the control.The American marten has become a focal species for the conservation of forested landscapes throughout North
3. Construct a 95% confidence interval to estimate the average difference in days taken for the marten to visit a chicken track box and a peanut butter track box. What do you conclude about the difference between the average number of days taken for the marten to visit each of these track boxes?The
4. Based on your answers to parts 1, 2, and 3, what can you conclude about the order of the effectiveness of the four different baits?The American marten has become a focal species for the conservation of forested landscapes throughout North America primarily due to its association with older
a. It is of interest to know if the sample data suggest the average blood glucose level is higher than 7.0 mmol/L. (i) State the appropriate hypotheses. (ii) Compute the test statistic for the hypotheses. (iii) (iv) Compute the approximate p-value associated with the test statistic in part (ii). Do
b. A diabetes researcher would like to study the variability in the fasting blood glu- cose level readings for the selected person in parta. The researcher believes that fasting blood glucose never varies more than 0.8 mmol/L. It is of interest to deter- mine if the sample data present sufficient
c. At a diabetic clinic, a group of people considered to have prediabetes (impaired fasting glucose) were randomly selected. Then they were randomly assigned in two groups, namely treatment group (subject to a strict diet plan) and placebo (normal diet), for a specified number of days. After the
d. The researcher was not too sure about the assumption of equality of variances and was under the impression that the population variance of the placebo group is greater than the variance of the treatment group. (i) Do the sample variances present sufficient evidence to indicate that the
e. Ten adults with symptoms of diabetes were randomly selected and a random blood glucose test was administered before and after a medication. Their blood glucose level readings were gathered before and after the medication; and the results are shown below.(i) Are the samples independent? Justify
You wish to show that the average hourly wage of clerical workers in the province of New Brunswick is different from $14, which is the national average. This is the alterna- tive hypothesis, written as H: 14 The null hypothesis is Hop=14 You would like to reject the null hypothesis, thus concluding
A milling process currently produces an average of 3% defectives. You are interested in showing that a simple adjustment on a machine will decrease p, the proportion of defectives produced in the milling process. Thus, the alternative hypothesis is H: p
According to Statistics Canada the mean birth weight of newborn babies in Canada is 3.372 kg for both sexes. However, a pediatrician in Alberta believes that mean birth of newborn babies for both sexes in Alberta is much higher than the national average. In this case, the alternative hypothesis is
Based on the online survey of a representative national sample of 1,006 Canadian adults, 54% respondents believe Canada should be a melting pot. A researcher thinks that the melting pot is particularly attractive for Quebecers and the proportion of Que- becers who support the idea of melting pot is
For the test of hypothesis in Example 9.1, the average hourly wage x for a random sample of 100 New Brunswick clerical workers might provide a good test statistic for testing Ho: 14 versus H: 14 If the null hypothesis Ho is true, then the sample mean should not be too far from the population mean =
The average weekly earnings for women in managerial and professional positions is $670. Do men in the same positions have average weekly earnings that are higher than those for women? A random sample of n = 40 men in managerial and profes- sional positions showed x = $725 and s = $102. Test the
The daily yield for a local chemical plant has averaged 880 metric tons for the last several years. The quality control manager would like to know whether this average has changed in recent months. She randomly selects 50 days from the computer data- base and computes the average and standard
Refer to Example 9.7. The quality control manager wants to know whether the daily yield at a local chemical plant-which has averaged 880 metric tons for the last several years has changed in recent months. A random sample of 50 days gives an average yield of 871 metric tons with a standard
Standards set by government agencies indicate that Canadians should not exceed an average daily sodium intake of 3300 milligrams (mg). To find out whether Canadians are exceeding this limit, a sample of 100 Canadians is selected, and the mean and stan- dard deviation of daily sodium intake are
Refer to Example 9.7. Calculate and the power of the test (1-3) when u is actually equal to 870 metric tons.
9.1 Find the appropriate rejection regions for the large-sample test statistic z in these cases:a. A right-tailed test with a = 0.01b. A two-tailed test at the 5% significance level
9.2 Refer to Exercise 9.1. Suppose that the observed value of the test statistic was z = 2.16. For the rejection regions constructed in parts a and b of Exercise 9.1, draw the appropriate conclusion for the tests. If appropriate, give a measure of the reliability of your conclusion.
9.5 For the three tests given in Exercise 9.4, use the p-value to determine the significance of the results. Explain what "statistically significant" means in terms of rejecting or accepting Ho and H.
9.7 Refer to Exercise 9.6.a. Calculate the p-value for the test statistic in partd. b. Use the p-value to draw a conclusion at the 5% significance level.c. Compare the conclusion in part b with the conclusion reached in part d of Exercise 9.6. Are they the same?
9.8 Refer to Exercise 9.6. You want to test Ho 2.3 against H: > 2.3.a. Find the critical value of x used for rejecting Ho.b. Calculate B = P(accept Ho when = 2.4).c. Repeat the calculation of for = 2.3, 2.5, and 2.6.d. Use the values of from parts b and c to graph the power curve for the test.
9.10 Airline Occupancy Rates High airline occupancy rates on scheduled flights are essential to corporate profitability. Suppose a scheduled flight must average at least 60% occupancy in order to be profit- able, and an examination of the occupancy rate for 120 10:00 A.M. flights from Toronto to
9.11 Ground Beef The meat department of a local supermarket chain packages ground beef in trays of two sizes. The smaller tray is intended to hold 1 kilo- gram (kg) of meat. A random sample of 35 packages in the smaller meat tray produced weight measurements with an average of 1.01 kg and a
9.12 Invasive Species In a study of the pernicious giant hogweed, Jan Pergl and associates compared the density of these plants in two different sites within the Caucasus region of Russia. In its native area, the aver- age density was found to be 5 plants/m. In an invaded area in the Czech
9.13 Advertising at the Movies "Welcome to the new movie pre-show!" Before you can see the newly released movie you have just paid to see, you must sit through a variety of trivia slides, snack bar ads, paid product advertising, and movie trailers. Although the total barrage of advertising may last
9.14 Potency of an Antibiotic A drug manufac- turer claimed that the mean potency of one of its antibi- otics was 80%. A random sample of n = 100 capsules were tested and produced a sample mean of x=79.7%, with a standard deviation of s = 0.8%. Do the data present sufficient evidence to refute the
9.15 Flextime Many companies are becoming involved in flextime, in which a worker schedules his or her own work hours or compresses work weeks. A com- pany that was contemplating the installation of a flex- time schedule estimated that it needed a minimum mean of 7 hours per day per assembly worker
9.16 Raise Your MCAT Test Scores! There are many books and crash courses available to prepare for the MCAT examination. An organization that offers such courses claims the average score improvement for its crash-course participants is between 29 to 37 points. Are the claims made by this
9.17 What's Normal? What is normal, when it comes to people's body temperatures? A random sam- ple of 130 human body temperatures, provided by Allen Shoemaker in the Journal of Statistical Education, had a mean of 36.81 Celsius and a standard deviation of 0.73. Does the data indicate that the
9.18 Sports and Achilles Tendon Injuries Some sports that involve a significant amount of running, jumping, or hopping put participants at risk for Achilles tendinopathy (AT), an inflammation and thickening of the Achilles tendon. A study in The American Journal of Sports Medicine looked at the
To determine whether car ownership affects a student's academic achievement, two random samples of 100 male students were each drawn from the student body. The grade point average for the n = 100 non-owners of cars had an average and variance equal to x = 2.70 and s=0.36, while = 2.54 and s = 0.40
Construct a 95% confidence interval for the difference in average academic achieve- ments between car owners and non-owners. Using the confidence interval, can you con- clude that there is a difference in the population means for the two groups of students?
9.19 Independent random samples of 80 measure- ments were drawn from two quantitative populations, 1 and 2. Here is a summary of the sample data: a. If your research objective is to show that , is larger than 2, state the alternative and the null hypotheses that you would choose for a statistical
9.21 Suppose you wish to detect a difference between and (either or ) and, instead of running a two-tailed test using a = 0.05, you use the following test procedure. You wait until you have collected the sample data and have calculated and 2. If x, is larger than 2, you choose the alternative
9.22 Cure for the Common Cold? An experi- ment was planned to compare the mean time (in days) required to recover from a common cold for per- sons given a daily dose of 4 mg of vitamin C versus those who were not given a vitamin supplement. Suppose that 35 adults were randomly selected for each
9.23 Healthy Eating Canadians are becoming more conscious about the importance of good nutrition, and some researchers believe we may be altering our diets to include less red meat and more fruits and vegetables. To test the theory that the consumption of red meat has decreased over the last 10
9.24 Lead Levels in Drinking Water Analyses of drinking water samples for 100 homes in each of two different sections of a city gave the following means and standard deviations of lead levels (in parts per million): a. Calculate the test statistic and its p-value (observed significance level) to
9.25 Starting Salaries, again In an attempt to compare the starting salaries for university graduates who majored in education and the social sciences (see Exercise 8.52), random samples of 50 recent university graduates in each major were selected and the follow- ing information was
9.26 Hotel Costs In Exercise 8.19, we explored the average cost of lodging at three different hotel chains. We randomly select 50 billing statements from the computer databases of the Marriott, Westin, and Doubletree hotel chains, and record the nightly room rates. A portion of the sample data is
9.27 Hotel Costs || Refer to Exercise 9.26. The table below shows the sample data collected to compare the average room rates at the Westin and Doubletree hotel chains." a. Do the data provide sufficient evidence to indicate a difference in the average room rates for the Westin and the Doubletree
9.28 MMT in Gasoline The addition of MMT, a compound containing manganese (Mn), to gasoline as an octane enhancer has caused concern about human exposure to Mn because high intakes have been linked to serious health effects. In a study of ambient air concentrations of fine Mn, Wallace and Slonecker
9.29 Noise and Stress In Exercise 8.55, you compared the effect of stress in the form of noise on the ability to perform a simple task. Seventy subjects were divided into two groups; the first group of 30 subjects acted as a control, while the second group of 40 was the experimental group. Although
9.30 What's Normal II Of the 130 people in Exer- cise 9.17, 65 were female and 65 were male." The means and standard deviations of their temperatures are shown below.a. Use the p-value approach to test for a significant difference in the average temperatures for males versus females.b. Are the
Regardless of age, about 20% of Canadian adults participate in fitness activities at least twice a week. However, these fitness activities change as the people get older, and occasionally participants become non-participants as they age. In a local survey of 'An equivalent test statistic can be
9.31 A random sample of n = 1000 observations from a binomial population produced x = 279.a. If your research hypothesis is that p is less than 0.3, what should you choose for your alternative hypoth- esis? Your null hypothesis?b. What is the critical value that determines the rejec- tion region
9.34 R and M Entertainment On behalf of the Canadian Teachers Federation (CTF), Erin Research conducted a study on Canadian children and their experience with communication media. The research indicates that 75% of students say R-movies should be open to kids age 12 and up, but less than 50% say
9.35 Plant Genetics A peony plant with red petals was crossed with another plant having streaky petals. A geneticist states that 75% of the offspring result- ing from this cross will have red flowers. To test this claim, 100 seeds from this cross were collected and germinated and 58 plants had red
9.36 Early Detection of Breast Cancer Of those women who are diagnosed to have early-stage breast cancer, one-third eventually die of the disease. Sup- pose a community public health department instituted a screening program to provide for the early detection of breast cancer and to increase the
9.37 Sweet Potato Whitefly Suppose that 10% of the fields in a given agricultural area are infested with the sweet potato whitefly. One hundred fields in this area are randomly selected, and 25 are found to be infested with whitefly.a. Assuming that the experiment satisfies the condi- tions of the
9.38 Taste Testing In a head-to-head taste test of store-brand foods versus national brands, Consumer Reports found that it was hard to find a taste difference in the two.12 If the national brand is indeed better tast- ing than the store brand, it should be judged as better more than 50% of the
9.39 Avian Flu Avian flu is clearly on the radar of Canadians. Given the amount of media coverage on the Avian Flu issue, it is not surprising to find that about 60% are concerned about it. However, the concern is fairly modest, only 19% being "very concerned," but this suggests that concern has
9.40 A Cure for Insomnia An experimenter has prepared a drug-dose level that he claims will induce sleep for at least 80% of people suffering from insom- nia. After examining the dosage, we feel that his claims regarding the effectiveness of his dosage are inflated.In an attempt to disprove his
9.41 Avian Flu, continued Refer to Exercise 9.39. The research report further indicates evidence that the public is somewhat concerned about whether or not Canadian health authorities are prepared to deal with this emerging problem. The survey showed that 45% believe that the Canadian health
9.42 Aboriginal People in Canada According to the most recent census data, Aboriginal people account for approximately 3.3% of the Canadian population. 14 However, the female-to-male ratio is slightly higher among Aboriginal peoples (51.2% female and 48.8% male) than in the total population (50.9%
9.43 Love for Pets in Canada Canadians own a total of 8 million dogs and cats. This total breaks down to approximately 3.5 million dogs and 4.5 million cats in the country, according to Statistics Canada. Half of all households have at least one pet, and pet ownership is higher in Alberta, Manitoba
The records of a hospital show that 52 men in a sample of 1000 men versus 23 women in a sample of 1000 women were admitted because of heart disease. Do these data pres- ent sufficient evidence to indicate a higher rate of heart disease among men admitted to the hospital? Use a = 0.05.
9.45 Refer to Exercise 9.43. Suppose, for practical reasons, you know that p, cannot be larger than p.a. Given this knowledge, what should you choose as the alternative hypothesis for your statistical test? The null hypothesis?b. Does your alternative hypothesis in part a imply a one- or two-tailed
9.47 Treatment versus Control An experiment was conducted to test the effect of a new drug on a viral infection. The infection was induced in 100 mice, and the mice were randomly split into two groups of 50. The first group, the control group, received no treatment for the infection. The second
9.48 Tai Chi and Fibromyalgia A new study (Exercise 7.13) indicates that tai chi, an ancient Chi- nese practice of exercise and meditation, may relieve symptoms of chronic painful fibromyalgia. The study assigned 66 fibromyalgia patients to take either a 12-week tai chi class (n = 33), or attend a
9.49 Movie Marketing Marketing to targeted age groups has become a standard method of advertising, even in movie theatre advertising. Advertisers use com- puter software to track the demographics of moviegoers and then decide on the type of products to advertise before a particular movie. 17 One
9.50 M&Ms In Exercise 8.60, you investigated whether Mars, Inc., uses the same proportion of red M&Ms in its plain and peanut varieties. Random sam- ples of plain and peanut M&Ms provide the following sample data for the experiment:Use a test of hypothesis to determine whether there is
9.51 Hormone Therapy and Alzheimer's Disease In the last few years, many research stud- ies have shown that the purported benefits of hormone replacement therapy (HRT) do not exist, and in fact, that hormone replacement therapy actually increases the risk of several serious diseases. A four-year
9.52 HRT, continued Refer to Exercise 9.51. Calculate a 99% lower one-sided confidence bound for the difference in the risk of dementia for women using hormone replacement therapy versus those who do not. Would this difference be of practical importance to a woman considering HRT? Explain.
9.53 Clopidogrel and Aspirin A large study was con- ducted to test the effectiveness of clopidogrel in combina- tion with aspirin in warding off heart attacks and strokes. 19 The trial involved more than 15,500 people 45 years of age or older from 32 countries. These people had been diagnosed with
9.54a. Define a and for a statistical test of hypothesis.b. For a fixed sample size n, if the value of a is decreased, what is the effect on ?c. In order to decrease both a and for a particular alternative value of p, how must the sample size change?
9.55 What is the p-value for a test of hypothesis? How is it calculated for a large-sample test?
9.56 What conditions must be met so that the z test can be used to test a hypothesis concerning a population mean?
9.57 Define the power of a statistical test. As the alternative value of gets farther from , how is the power affected?
9.58 Acidity in Rainfall Refer to Exercise 8.34 and the collection of water samples to estimate the mean acidity (in pH) of rainfalls in Eastern Canada. As noted, the pH for pure rain falling through clean air is approximately 5.7. The sample of n = 40 rainfalls produced pH readings with 3.7 and
9.59 Washing Machine Colours A manufacturer of automatic washers provides a particular model in one of three colours. Of the first 1000 washers sold, it is noted that 400 were of the first colour. Can you con- clude that more than one-third of all customers have a preference for the first colour?a.
9.60 Commercials in Space The commercialism of our space program20 was the topic of Exercise 8.67. In a survey of 500 men and 500 women, 20% of the men and 26% of the women responded that space should remain commercial-free.a. Is there a significant difference in the population proportions of men
9.61 Bass Fishing The pH factor is a measure of the acidity or alkalinity of water. A reading of 7.0 is neutral; values in excess of 7.0 indicate alkalinity; those below 7.0 imply acidity. Loren Hill states that the best chance of catching bass occurs when the pH of the water is in the range 7.5 to
9.62 Traffic Tickets An Ontario website (www.trafficticket.com) claims "We fully dismiss [at least] 94% of all cases."a. Suppose that we examine 25 cases and exactly 15 were fully dismissed. Would you reject the claim of 94% or more made by the website?b. Suppose that the records for the past year
9.63 White-Tailed Deer In an article entitled "A Strategy for Big Bucks," Charles Dickey discusses studies of the habits of white-tailed deer that indicate that they live and feed within very limited ranges- approximately 607,028 to 829,605 square metres (m). 22 To determine whether there was a
9.64 Female Models In a study to assess various effects of using a female model in automobile adver- tising, 100 men were shown photographs of two auto- mobiles matched for price, colour, and size, but of different makes. One of the automobiles was shown with a female model to 50 of the men (group
9.65 Bolts Random samples of 200 bolts manufac- tured by a type A machine and 200 bolts manufactured by a type B machine showed 16 and 8 defective bolts, respectively. Do these data present sufficient evidence to suggest a difference in the performance of the machine types? Use a = 0.05.
9.66 Biomass Exercise 7.74 reported that the bio- mass for tropical woodlands, thought to be about 35 kilograms per square metre (kg/m), may in fact be too high and that tropical biomass values vary region- ally-from about 5 to 55 kg/m.23 Suppose you mea- sure the tropical biomass in 400 randomly
9.67 Anti-Terrorism Bill C-36 The Anti-Terrorism Bill was passed by the House of Commons on November 28, 2001. Many people argued that the law increased burden for charities. A researcher believed that the fraction p of Conservatives in favour of the Anti-Terrorism Bill was greater than fraction p
9.68 Anti-Terrorism Bill C-36, continued Refer to Exercise 9.67. Some thought should have been given to designing a test for which is tolerably low when p exceeds p2 by an important amount. For example, find a common sample size n for a test = with a 0.05 and 0.20, when in fact p exceeds P2 by 0.1.
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