Question: Loss of calcium is a serious problem
Loss of calcium is a serious problem for older women. To investigate the amount of loss, a researcher measured the initial amount of bone mineral content in the radius bone of the dominant hand of elderly women and then the amount remaining after one year. The differences, representing the loss of bone mineral content, are given in the following table (courtesy of E. Smith).
(a) Find the sample mean.
(b) Does the sample mean or the median give a better indication of the amount of mineral loss?
> Charles Darwin performed an experiment to determine if self-fertilized and cross-fertilized plants have different growth rates. Pairs of Zea mays plants, one self- and the other cross-fertilized, were planted in pots, and their heights were measured afte
> Independent samples of sizes nA = 2 and nB = 5 are taken from two continuous populations. (a) Enumerate all possible collections of ranks associated with population A. Also attach probabilities to these rank collections assuming that the populations are
> We presented data on the blood pressure of 15 persons before and after they took a pill. (a) Perform a sign test, with a near .05, to determine if blood pressure has decreased after taking the pill. (b) Perform a Wilcoxon signed-rank test to det ermine
> A married couple's monthly credit charges are divided into his and hers and the difference, husband's minus wife's, is calculated. A random sample of 30 married couples yielded the Wilcoxon signed-rank statistic T+ = 325. What is the significance probabi
> The null distribution of the Wilcoxon signed-rank statistic T+ is det ermined from the fact that under the null hypothesis of a symmetric distribution about zero, each of the ranks 1, 2, ... , n is equally likely to be associated with a positive sign or
> The following measurements of the diameters (in feet) of Indian mounds in southern Wisconsin were gathered by examining reports in the Wisconsin Archaeologist (courtesy of J. Williams). 22 24 24 30 22 20 28 30 24 34 36 15 37 (a) Create a dot
> Referring to Exercise 15.13, apply the Wilcoxon signed-rank test with a near .05. Data from Exercise 15.13: Two critics rate the service at six award-winning restaurants on a continuous 0 to 10 scale. Is there a difference between the critics' ratings?
> Use Appendix B, Table 9, to find: (a) P ( T+ ≥ 65 ] when n = 12. (b) P ( T+ ≤ 10] when n = 10. (c) The value c such that P [ T+ ≥ c ] = .039 when n = 8. (d) The values c1 and c2 such that P ( T+ &
> A social researcher interviews 25 newly married couples. Each husband and wife are independently asked the question: "How m any children would you like to have?" The following data are obtained. Do the data show a significant difference of opinion betwee
> Two critics rate the service at six award-winning restaurants on a continuous 0 to 10 scale. Is there a difference between the critics' ratings? (a) Use the sign test with a below .05. (b) Find the significance probability.
> In a taste test of two chocolate chip cookie recipes, 13 out of 17 subjects favored recipe A. Using the sign test, find the significance probability when H1 states that recipe A is preferable.
> One aspect of a study of gender differences involves the play behavior of monkeys during the first year of life ( courtesy of H. Harlow, U. W. Primate Laboratory). Six male and six female monkeys are observed in groups of four families during several ten
> If nA = 1 and nB = 14, find (a) The rank configuration that most strongly supports H1 : Population A is shifted to the right of population B. (b) The null probability of WA = 15. (c) Is it possible to have a = .05 with these sample sizes?
> Independent random samples of sizes nA = 4 and nB = 2 are taken from two continuous populations. (a) Enumerate all possible collections of ranks associated with the smaller sample in the combined sample ranking. Attach probabilities to these rank collect
> Using Appendix B, Table 7 , find the upper 10% point of the F for (a) d.f. = (4, 5) (b) d.f. = (4, 10) (c) d.f. = (4, 15) (d) d.f. = (4, 30) (e) Use software to verify your answers. (f) What effect does increasing the denominator d.f. have on the percen
> Using Appendix B, Table 7 for percentage points for the F distribution, find (a) The upper 5% point when v1 = 5 and v2 = 12. (b) The upper 5% point when v1 = 12 and v2 = 5. ( c) Use software to verify your answers to Part a and Part b.
> If you were to use the data on the length of major league baseball games in Exercise 2.43 to estimate the total amount of digital memory needed to film another 10 major league baseball games, which is the more meaningful description, the sample mean or t
> Reading levels vary between different magazines. To avoid difficulties caused by different typefaces and sizes, an investigator just count ed the number of letters and punctuation signs. Random samples of 20 sentences were selected from The New Yorker, S
> The abilities of six different brand athletic socks to wick moisture are rated. The rating is based on the time to reach l % moisture content after being subjected to a gentle mist. Use the relations for sums of squares and d.f. to complete the following
> An investigator wonders how the frequency of checking a smart phone varies with current activity. Students are assigned to one of four conditions: listening to a lecture, studying, eating a meal, or on a date. The response they record is the number of ti
> Refer to Exercise 14.28. Present the ANOVA table. What conclusions can you draw from th e two F tests? Take  = .05 . Data from Exercise 14.28: The yield, in pounds, of three types of heritage tomatoes (treatments) is obtained after pl
> Since restaurant C had unacceptable scores, it was decided to concentrate on the other restaurants. Five additional students are assigned to these restaurants. Suppose the quality scores, for all 11 visits combined, are Repeat Exercise 14.2 using these d
> Three loaves of bread, each made according to a different recipe, are baked in one oven at the same time. Because of possible uncontrolled variations in oven performance, each baking is treated as a block. This procedure is repeat ed five times, and the
> The yield, in pounds, of three types of heritage tomatoes (treatments) is obtained after planting them in three equal sized plots within a site (block). Types are randomly assigned to plots. A total of four different sites are used. Suppose the data are
> Refer to Exercise 14.25. Present the ANOVA table. What conclusions can you draw from the two F tests? Take  = .05. Data from Exercise 14.25: The hours of relief are measured under a placebo, Brand A, and Brand B sinus medicines. Call
> Suppose you wish to compare three different brands of tick collars for dogs. You have available three of each of the breeds Poodle, Lab, Collie, and Dachshund. Explain h ow you would assign a brand of tick collar to each of the 12 dogs in order to conduc
> On opening day one season, 10 major league baseball games were played and they lasted the following numbers of minutes. 167 211 187 176 170 158 198 218 145 232 Find the sample median.
> The hours of relief are measured under a placebo, Brand A, and Brand B sinus medicines. Call these treatment 1, treatment 2, and treatment 3, respectively. They are each given in succession, in random order, to each subject with two days between each tre
> Refer to the MINITAB commands but also check the box Store residuals. Graph the residuals from an ANOVA of the iris data. (a) In a normal-scores plot using all of the residuals. (b) Individual normal-scores plots for each type of iris.
> Referring to the density of bread data in Exercise 14.14, plot the residuals (a) In a dot plot using all of the residuals. (b) Individual dot plots for each recipe. Data from Exercise 14.14: Three bread recipes are to be compared with respect to densit
> Referring to the sound distortion data in Example 1, graph the residuals in (a) A dot plot using all of the residuals. (b) Individual dot plots for each model of earbuds.
> Determine the expression for the length of the t interval for µ1 - µ2 and the multiple-t interval for µ1 - µ2 when m = 10. The ratio of lengths does not depend on the data. Evaluate this ratio for a = .10 and n - k = 15.
> A study was conducted to examine the degree that the length of gestation influences school achievement for children born in the normal range of 3 7 to 41 weeks gestation. Academic achievement was measured by a third grade reading test that we summarize f
> Subjects must press a button when they h ear a signal. The three treatments are three different intensities of the signal. The time elapsed between presentation of the signal and when the button is pushed, are recorded in hundredths of a second. Suppose
> As suggested by the photo in the beginning of this chapter, consumers can rate new HDTVs. Many Web sites use a five point scale. Although the individual responses are not normal, the central limit result applied to each of the four samples does justify t
> Construct the 90% multiple-t confidence intervals using the sound distortion data in Example 1.
> Taking a = .OS and n - k = 26, determine the appropriate percentile of the t distribution when calculating the multiple-t confidence intervals with (a) m = 3 and (b)m = 5.
> Physical education researchers interested in the development of the overarm throw measured the horizontal velocity of a thrown ball at the time of release. The results for first-grade children (in feet/second) (courtesy of L. Halverson and M. Roberton) a
> Refer to the data on reading levels in Exercise 14.7. Test for equality of means. Take a = .05. Data from Exercise 14.7: Reading levels vary between different magazines. To avoid difficulties caused by different typefaces and sizes, an investigator just
> Test for equality of means based on the travel expense data in Exercise 14.6. Take  = .05. Data from Exercise 14.6: Referring to the lying experiment. The subjects were also able to cheat on the tax form by claiming excessive travel e
> Three bread recipes are to be compared with respect to density of the loaf. Five loaves will be baked using each recipe. (a) If one loaf is made and baked at a time, how would you select the order? (b) Given the following data, conduct an F test for equa
> Test for equality of means based on the data in Exercise 14.2. Take  = .05. Data from Exercise 14.2: Subjects must press a button when they h ear a signal. The three treatments are three different intensities of the signal. The time e
> Using the data from Exercise 14.1, test for equality of means using a = .05. Data from Exercise 14.1: A group of eight students want to compare the quality of food at 4 campus area restaurants. The two students randomly assigned to a site, dine on diff
> Based on the current General Social Survey, an index is created from the few questions asking about the degree of confidence in government. Six age groups are compared. Suppose the values of the index result in the following ANOVA table. Carry out the F
> A psychologist investigating the connection between music and memory, randomly assigns subjects to one of three treatment groups. The first group hears white noise, the second Mozart, and the third heavy metal. With the appropriate background sound turne
> A group of eight students want to compare the quality of food at 4 campus area restaurants. The two students randomly assigned to a site, dine on different nights. Each student rates the quality of their meal on a ten point scale. Calling the restaurant
> (a) By expanding the square on the right-hand side of show that the x2 statistic can also be expressed as (b) Express the x2 statistic as where P^i = ni / n is the observed relative frequency. Note that doubling the sample size, while maintaining the sam
> Observations of 80 litters, each containing 3 rabbits, reveal the following frequency distribution of the number of male rabbits per litter. Under the model of Bernoulli trials for the sex of rabbits, the probability distribution of the number of males p
> Internet consumer ratings of the previous model smart speaker, based on ab out 100,000 ratings, can be summarized as: The ratings of the new model by 200 persons yielded the following frequency table. Do these data show a significantly different distribu
> Cross-fertilizing a pure strain of red flowers with a pure strain of white flowers produces pink hybrids that have one gene of each type. Crossing these hybrids can lead to any one of four possible gene pairs. Under Mendel's theory, these four are equall
> A shipment of assorted nuts is labeled as having 45% walnuts, 20% hazelnuts, 20% almonds, and 15% pistachios. By randomly picking several scoops of nuts from this shipment, an inspector finds the counts shown. Could these findings be a strong basis for a
> A market researcher wishes to assess consumers' preference among three different colors available on name-brand dishwashers. The following frequencies were observed from a random sample of 150 recent sales. Test the null hypothesis, at a = .05, that all
> Referring to the data in Exercise 13.2, test the null hypothesis that the probability of the blood types 0, A , B, and AB is in the ratios 4 :4 : l:l. Use  = .05. Data from Exercise 13.2: Recorded here is the frequency distribution of
> Do Exercise 13.24 on a computer. Data from Exercise 13.24: Out of 120 members of a club who responded to a survey, 80 said that the golfing facilities were influential, 53 said the dining facilities were influential, and 25 said both were influential in
> The analysis of a contingency table becomes tedious especially when the size of the table is large. Using a computer makes the task quite easy. The output is as follows: (a) Compare this output with the calculations. (b) Do Exercise 13.23 on a computer.
> Out of 120 members of a club who responded to a survey, 80 said that the golfing facilities were influential, 53 said the dining facilities were influential, and 25 said both were influential in their decision to join the club. (a) Construct a two-way t
> Lying can have major economic and social implications. A double blind trial was conducted to study the effect of testosterone on lying. Among 9 1 healthy men, n 1 = 46 received 50 mg of testosterone on a skin patch while the control group, n2 = 45, recei
> Refer to the mattress data in the chapter opener. (a) Can the observed differences be explained by chance or are mattress type and satisfaction level associated? (b) Determine the contribution of each cell to the x2 statistic. (c) For each cell, deter
> Old Faithful, the most famous geyser in Yellowstone Park, had the following durations (measured in seconds) in six consecutive eruptions: 240 248 113 268 117 253 (a) Find the sample median. (b) Find the sample mean.
> Th e article What Makes a Great Tweet concludes that only 36% of tweets are worth reading. A total of 4,220 tweets were rated in this study. Identify a statistical population and the sample.
> A consultant to all kinds of retailers suggests that they have plenty of baskets available for customers. He bases his suggestion on data collected by watching videos from hidden cameras. Suppose that out of 200 customers, 80 picked up a basket when they
> Applicants for public assistance are allowed an appeals process when they feel unfairly treated. At such a hearing, the applicant may choose self-representation or representation by an attorney. The appeal may result in an increase, decrease, or no chang
> Recorded here is the frequency distribution of the blood types of 100 persons who have volunteered to donate blood at a plasm a center. Test the goodness of fit of the model, which assumes that the four blood types are equally likely in the population of
> With reference to the sleep data in Table D. l 0 of the Data Bank, make two categories of snorers on the basis of the last variable: those who responded three or more times a week so their score is coded 4 or 5 and those who responded less than three tim
> Using the computer. The analysis of a contingency table can be conveniently done on a computer. For an illustration, we present here a MINITAB analysis of the data in Table 6, Example 5. The output is as follows: (a) Compare this output with the calculat
> Osteoporosis, or a loss of bone minerals, is a common cause of broken bones in the elderly. A researcher on aging conjectures that bone mineral loss can be reduced by regular physical therapy or certain kinds of physical activity. A study is conducted on
> Referring to Exercise 13.10, compare the conclusions of the level a = .OS (a) x2 test of homogeneity. (b) Z test that sales are higher with the modified page. Data from Exercise 13.10: A strategy called A/B testing is being implemented by many e-comme
> Using only the two conditions "signature at the top" and "no signature required, " (a) Make a 2 X 2 table and test H0 : PT = PN versus H1 : PT < PN at a = .02. (b) Obtain a 95% confidence interval for PN - PT·
> Using only the two conditions "signature at bottom" after filling out the form and "no signature required." (a) Conduct a x2 test. Take a = .05. (b) Does the conclusion of your test prove that the two probabilities PN and p8 are equal? Clearly explain t
> Referring to Exercise 3.10, double all of the counts. (a) Calculate the x2 statistic. (b) Compare your answer in Part a to the answer in Exercise 13.10 Data from Exercise 13.10: A strategy called A/B testing is being implemented by many e-commerce fir
> With reference to the number of returns, obtain the sample (a) mean and (b) median.
> Nausea from air sickness affects some travelers. In a comparative study of the effectiveness of two brands of motion sickness pills, brand A pills were given to 45 persons randomly selected from a group of 90 air travelers, while the other 45 persons wer
> Do Exercise 13.10 using computer software. Data from Exercise 13.10: A strategy called A/B testing is being implemented by many e-commerce firms. A product web page is modified by adding a picture or other visual change. Then, a fraction of the traffic
> A strategy called A/B testing is being implemented by many e-commerce firms. A product web page is modified by adding a picture or other visual change. Then, a fraction of the traffic is routed to the new page. For a sample of 1000 consumers who visit th
> Given below are the frequencies observed from 320 tosses of a die. Do these data cast doubt on the fairness of the die?
> Given the data, (a) Fit a quadratic curve using the equations. (b) Verify your answers using software.
> Referring to your answer to 12.7 p art 6, (a) Find a 95% confidence interval for βj . (b) Test H0 :p2 = 4 versus H1 :p2 > 4 with  = .05. Data from Exercise 12.7: Consider the data on a response variable and two predic
> Consider the data on a response variable and two predictor variables. (a) Fit a multiple regression model with response variable y. Note that the equations in the middle of page 391 simplify for these data. (b) Verify your answer using software.
> Consider the multiple linear regression model where βo = 2, β1, = -3, β2 = 2, and the normal random variable e h as standard deviation 3 . What is the mean of the response Y when x1 = 3 and x2 = - 2?
> In an experiment ( courtesy of W. Burkholder) involving stored-product beetles (Trogoderma glabrum) and their sex-attractant pheromone, the pheromone is placed in a pit-trap in the centers of identical square arenas. Marked beetles are then released alon
> An experiment was conducted for the purpose of studying the effect of temperature on the life length of an electrical insulation. Specimens of the insulation were tested under fixed temperatures, and their times to failure recorded. (a) Fit a straight li
> With reference to the extracurricular activities data in Exercise 2.3, obtain the (a) Sample mean. (b) Sample median. (c) Comment on the effect of a few large observations. Data from Exercise 2.3: A student at the University of Wisconsin surveyed 40
> Referring to Example 1, Table 1, three additional trials are now available. (a) Fit a straight line model with the square root of stopping distance as the response variable. (b) Find the proportion of variation explained by x. (c) Verify your answer to
> Refer to the data of Exercise 12.1. (a) Consider the reciprocal transformation y' = 1 / y and plot the scatter diagram of y' versus x. (b) Fit a straight line regression to the transformed data. (c) Calculate r 2 and comment on the adequacy of the fit.
> Referring to Exercise 12 .11 , we have added one more predictor x3 = skull width (cm). The output for a regression analysis. (a) Test H0 : β3 = 0 versus H0 : β3 ≠ 0 with = .05. (b) Test H0 : β2 = 0 versus H0 : β2 ≠ 0 with = .05. (c) Estimate the expe
> Referring to Example of the Data Bank, we now use two predictors, x1 = girth (cm) and x2 = length to predict y = weight (lb) of a polar bear. The output from a regression analysis. (a) How m any polar bears were included in the analysis? (b) Identify th
> Consider the data on all of the wolves in Table D.9 of the Data Bank concerning age (years) and canine length (mm). (a) Obtain the least squares fit of the straight line regression model Y = βo + β1x + e to predict canine length from age. (b) Obtain the
> Developers have built a small robotic vehicle that can travel over rough terrain. They recorded the time y, in minutes, that it takes to travel a fixed distance over various but similar terrains. For a fixed run, the robot's motor is set at a nominal spe
> Consider the linear regression model where β0 = -2, β1 = -1, and the normal random variable e has standard deviation 3. (a) What is the mean of the response Y when x = 3 ? When X = 5? (b) Will the response at x = 3 always be lar
> Graph the straight line for the means of a linear regression model Y = β0 + β1x + e having β0 = 7 and β1 = 2.
> Graph the straight line for the means of the linear regression model having β0 = 3, β1 = -2.
> Under the linear regression model: (a) Determine the mean and standard deviation of Y, for x = 1, when β0 = 3, β1 = -2, and ( = 3. (b) Repeat part(a) with x = 2.
> The weights (oz) of nineteen babies born in Madison, Wisconsin, are summarized in the computer output Locate two measures of center tendency, or location, and interpret the values.
> Under the linear regression model: (a) Determine the mean and standard deviation of Y, for x = 3, when βo = 1, β1 = 5, and ( = 4. (b) Repeat part(a) with x = 2.
> Show that the SS due to regression, S2xy / Sxx, can also be expressed as β^21 Sxx.
> (a) Show that the sample correlation coefficient rand the slope β^1 of the fitted regression line are related as (b) Show that SSE = (1 - r2) Syy·
> Refer to Exercise 11.30. According to the computer output in Table 8, find the proportion of y variability explained by x. Data from Exercise 11.30: According to the computer output in Table 8: (a) What model is fitted7 (b) Test, with a = .05, if the
> Identify the values of the parameters βo, β1, and (; in the statistical model Y = 6 - 3x + e where e is a normal random variable with mean O and standard deviation 3.
> Refer to Exercise 11.28. According to the computer output in Table 7, find the proportion of y variability explained by x. Also, calculate R2 from the analysis of variance table. Data from Exercise 11.28: According to the computer output in Table 7: (a
> Refer to Exercise 11.25. (a) What proportion of y variability is explained by the linear regression on x? (b) Find the sample correlation coefficient. Data from Exercise 11.25: An engineer found that by adding small amounts of a compound to rechargeabl
