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statistics principles and methods

Statistics Principles And Methods 8th Edition Richard A. Johnson, Gouri K. Bhattacharyya - Solutions

Describe a procedure, based on 12 identical cards, that generates an observation from the probability distributionSpecify appropriate numbers, one for each card, that make this probability distribution prevail for the number appearing on a randomly drawn card. x -1 0 012 f(x) 2/12 3/12 4/12 3/12
Probabilities, provided by the engaged couple, can help control the number of guests at their wedding. At the planning stage, decisions are necessarily based on the number of persons expected to respond yes to the RSVP. Concerning one group of four adults, the couple assigns probabilities to the
Among six new apps available this month, only three will prove to be useful to you in the long run. Suppose you purchase three of the apps at random. Find the probability distribution of X, the number of useful apps you select.
Suppose the probability distribution of a random variable X has the distributionCalculate the mean and standard deviation of this distribution. f(x) 60 1 77 x for x = 2, 3, 4, 5
In bidding for a contract to cater a large wedding, a caterer expects she will have a net profit of $3000 if she wins the contract and net loss of $60 if her bid fails. If the probability of getting the contract is .3, calculate her expected return.
A jar contains 20 pieces of wrapped hard candies, 7 are raspberry, 10 are butterscotch, and 3 are other flavors. Consider 4 successive draws of l piece of candy taken at random from the jar. Suppose the selection of butterscotch is the event of interest. For each of the following situations, state
Refer to Exercise 5.43 but instead assume that the jar contains 2000 pieces of wrapped hard candies, of which 700 are raspberry, 1000 are butterscotch , and 300 are other flavors. Repeat parts (a)-(c) of Exercise 5.43 in this setting.Data from in Exercise 5.43A jar contains 20 pieces of wrapped
A backpacking party carries three emergency signal flares, each of which lights with a probability of .94. Assuming that the flares operate independently, find:(a) The probability that at least one flare lights.(b) The probability that exactly two flares light.
According to a recent survey, about 40% of adults reported that the most helpful method to achieve a goal is to share their goal with friends and ask them for support. Let S be the event that an adult feels sharing with friends is the most helpful method to achieve a goal. If a random sample of
According to the U.S. Census Bureau, about 10% of persons between 25 and 29 years old live alone. For a random sample of size n, use the binomial table, or software, to find the probability of(a) 1 or fewer persons living alone when n = 14.(b) 2 or more persons living alone when n =14.(c) 2
Referring to Exercise 5.71, find:(a) The expected number of college seniors, in a random sample of 20, supporting the increased funding.(b) The probability that the number of sampled college seniors supporting the increased funding equals the expected number.Data from in Exercise 5.71Suppose that
According to a recent Federal Trade Commission Report, about .13% of the persons in the 20-29 year old age group filed identity theft complaints. From a random sample of n = 600 persons 20-29 years old, let X denote the number who filed an identity theft complaint last year. Assuming that the
John and two of his friends have one-on-one basketball tournaments almost every week. Each person plays each other p erson once. Let X be the number of games that John wins next week X. Based on the results of the games over the past year, John constructs the following probability distribution for
Refer to Exercise 5.88 but now suppose the sampling is done in two stages: First a list is selected at random and then, from that list, two persons are selected at random without replacement. Let Y denote the number of females in the sample.(a) List the elementary outcomes concerning the possible
A student buys a lottery ticket for $2. The prize is a bicycle worth $500. The winning ticket will be chosen at random from 2000 tickets that were sold.(a) What is the probability that the student will win a bicycle?(b) Determine the student's expected gain.
Determine the following probabilities from the curve ∫(x) diagrammed in Exercise 6.1 (a).(a) P( 0 < X < .2)(b) P(1.8 < X < 2.0)(c) P(.4 < X < .6)Data from in Exercise 6.1Which of the functions sketched below could be a probability density function for a continuous random
If in three Bernoulli trials the probability that the first two trials are both failures is ¼, what is the probability that the first two are successes and the third is a failure?
The proportion of people having the blood type O-positive in a large southern city is .38. For two randomly selected donors:(a) Find the probability of at least one type 0 .(b) Find the expected number of type 0 .(c) Repeat parts (a) and (b) if there are three donors .
Find the standardized variable Z if X has(a) Mean 13 and standard deviation 3.(b) Mean 57 and standard deviation 16.(c) Mean 142 and variance 36.
Two pieces of wood need to be glued together. After applying glue to both pieces, they must be clamped to obtain the best results. The required clamping time in any application is a random variable. Find the standardized variable Z, for the clamping time X, when using a manufacturers'(a) Wood glue
Find the area under the standard normal curve to the left of(a) z = 1.37 (b) z = .34(c) z = -1.61 (d) z = -2.53
Find the area under the standard normal curve to the left of(a) z = 1.63(b) z = -1.07(c) z = -1.07(d) z = -1.55
Find the area under the standard normal curve to the right of(a) z = 1.27(b) z = .54(c) z = -1 .63(d) z = -1.325 (interpolate)
Find the area under the standard normal curve to the right of(a) z = .63(b) z = 2.63(c) z = -1.23(d) z = 1.635 (interpolate)
Find the area under the standard normal curve over the interval(a) z = -.85 to z = .85(b) z = -1.15 to z = 1.15(c) z = .58 to z = 2.03(d) z = -.845 to z = 1.366 (interpolate)(e) Check all of your answers using software.
For a standard normal variable, Z, find(a) P(Z < .73)(b) P(Z < -.73)(c) P(Z > 1.42)(d) P(Z > -1.42)(e) P(-1.4 < Z < 2.57)(f) P( .07 < Z < .70 )(g) P(-1.65 < Z < -.45)(h) P(IZI < 1.96)
For X having the density in Exercise 6.55. Find (a) P(X > .8) (b) P(.4 ≤ X ≤ .8) (c) P(.4 < X < .8).Data from in Exercise 6.55Determine (a) the median and (b) the quartiles for the distribution shown in the following illustration. f(x) 1 0 1 x
Find(a) P(Z < .14)(b) The 14th percentile of Z(c) P(Z < .86)(d) The 86th percentile of Z.
Let the number of successes X have a binomial distribution with n = 20 and p = .5(a) Find the exact probability of each of the following:X = 11 6 ≤ X ≤ 12 6 < X < 12(b) Apply the normal approximat ion to each situation in part (a).
The bell-shaped histogram for the heights of three-year-old red pine seedlings on shown below is consistent with the assumption of a normal distribution having mean = 280 and sd = 58 millimeters. Let X denote the h eight, at three years of age, of the next red pine that will be measured. Find(a)
If X has a normal distribution with μ = 90 and σ = 4, find b such that(a) P(X < b) = .6700(b) P(X > b) = .0110(c) P(IX - 90 | < b) = .966(d) Check all of your answers using software.
The population density function and that for the sampling distribution of X̅, for n = 2, are shown in Figure 6. Identify which one is the sampling distribution and explain your answer. 24 1 I 0 Figure 6 Two density functions. Exercise 7.11. 1 2
A random sample of size 2 is selected, with replacement, from the set of numbers { 0, 2, 6 }.(a) List all possible samples and evaluate X̅ and s2 for each .(b) Determine the sampling distribution of X̅.
Refer to the data on earthquakes of magnitude greater than 6.5 in Exercise 2.20. The data suggests that one plausible model, for X = magnitude, is a population distribution having mean μ = 7 .05 and standard deviation σ = .43. Calculate the expected value and standard deviation of X̅ for a
A bride-to-be asks a prospective wedding photographer to show a sample of her work. She provides ten pictures. Should the bride-to-be consider this a random sample of the quality of pictures she will get? Comment.
A population has distributionLet X1 and X2 be independent and each have the same distribution as the population.(a) Determine the missing elements in the table for the sampling distribution of X̅ = (X1 + X2 ) /2.(b) Find the expected value of X̅.(c) If the sample size is increased to 25, give the
Using the sampling distribution determined for X̅ = (X1 + X2) / 2 in Exercise 7.4, verify that E [X̅] = μ and sd(X̅) = σ/ √2.Data from in Exercise 7.4From the set of numbers { 3, 5, 7 }, a random sample of size 2 is selected with replacement.
The weekly flier from a large box store always features many items having a mail-in rebate. A random sample of 50 submissions produced the summary statistics X̅ = 43 and s = 7. 7 days for the time interval between submission and receipt of the rebate check. Construct a point estimate of the
When estimating μ from a large sample, suppose that one has found the 95% error margin of X̅ to be 10.4. From this information, determine:(a) The estimated S.E. of X̅.(b) The 90% error margin.
A company wants to check the consistency of electronic copies of signatures for consumer credit purchases. A sample of 49 electronic signatures are available from the same customer. One measure of consistency in signing is the total length that the script is outside the signature box. The sample of
A nutritionist records the time to eat an evening meal, at a student cafeteria, for 64 students. She obtains X̅ = 36.2 and s = 5.6 minutes and reports that the 95% confidence interval for the mean time to eat isor (30.79, 41.61)(a) Is this statement correct?(b) Does the interval (30 .79, 41.6 1)
In a study designed to find differences in personality between cat and dog lovers, 3 subjects were first classified according to their answers to a questionnaire about attitudes towards pets, and previous pet ownership. The n = 66 subjects classified as dog lovers were then scored on the basis of
The weekly flier from a large box store always features many items having a mail-in rebate. A random sample of 50 submissions produced the summary statistics X̅ = 43 and s = 7.7 days for the time interval between submission and receipt of the rebate check. Construct a 95% confidence interval for
In a study to determine whether a certain stimulant produces hyperactivity, 44 mice were injected with 10 micrograms of the stimulant. Afterward, each mouse was given a hyperactivity rating score. The mean score was X̅ = 15.3 and s = 2.4. Give a 95% confidence interval for the population mean
The weekly flier from a large box store always features many items having a mail-in rebate. A random sample of 50 submissions produced the summary statistics X̅ = 43 and s = 7. 7 days for the time interval between submission and receipt of the rebate check.(a) Is the claim that μ > 40 days
A national safety council wishes to estimate the proportion of automobile accidents that involve pedestrians. How large a sample of accident records must be examined to be 96% certain that the estimate does not differ from the true proportion by more than .04?
California Condors, birds with up to a nine foot wingspan, almost became extinct over 20 years ago. Then a program began where birds are raised in captivity and and later released into the wild. According to the the U.S. Fish and Wildlife Service, 157 out of 166 birds survived over a recent year.
Given here are the observed sample proportions p̂ in the contexts of parts (a-d.) of Exercise 8.62. Calculate the test statistic and draw a conclusion of the test at the specified level of significance.(a) p̂ = .233(b) p̂ = .72(c) p̂ = .54(d) Claim p̂ = .86Data from in Exercise 8.62Each part
From telephone interviews with 1032 adults, it was found that 68% of those persons supported tougher legislation for antipollution measures. Does this poll substantiate the conjecture that more than 65% of the adult population are in favor of tougher legislation for antipollution measures?
Use software to find(a) T < -1.729 when d.f. = 19.(b) ITI > 2.179 when d.f. = 12.(c) -1.703 < T < 1.703 when d.f. = 27.(d) -1.833 < T < 2.821 when d.f. = 9.
Use software to find the number b so that(a) P[T < b] = .95 when d.f. = 5.(b) P[-b < T < b] = .95 when d.f. = 37.(c) P[T > b] = .01 when d.f. = 2.(d) P[T > b] = .99 when d.f. = 13.
Use software to find the probability(a) T ≤ –1.3 when d.f. = 12.(b) T > l.7 when d.f. = 21.(c) ITI > l.35 when d.f. = 33.(d) Find c such that P[T > c] = .015 when d.f. = 18.(e) Find c such that P[I T l > c] = .04 when d.f. = 42.
The monthly rent (dollars) for a two-bedroom apartment on the west side of town was recorded for a sample of ten apartments.1249 1590 1650 1625 1345 1725 1620 1200 1020 1200Obtain a 95% confidence interval for the mean monthly rent for
Given the sample 4 7 5 4(a) Evaluate the t-statistic for testing H0 : μ0 = 3.(b) Use software to check your value of t.R: t.test(x,mu = 3) after x= c( 4, 7, 5, 4)
Refer to Exercise 9.12. Do these data support the claim that the mean monthly rent for a two-bedroom apartment differs from 1200 dollars? Take a = .05.Data from in Exercise 9.12The monthly rent (dollars) for a two-bedroom apartment on the west side of town was recorded for a sample of ten
Refer to Exercise 9 .10, where a zoologist collected 20 wild lizards in the southwestern United States. Do these data substantiate a claim that the mean length is greater than 128 mm? Test with a = .05.Data from in Exercise 9 .10A zoologist collected 20 wild lizards in the southwestern United
Ten participants tried a new weight loss diet for two months. The resulting weight losses (pounds} are20.2 11.0 15.4 9 .9 7.9 15.6 15.5 23.1 11.0 11.4(a} Construct a 95% confidence interval for the population mean amount μ of decrease
Refer to Exercise 9.41. Construct a 95% confidence interval for the true standard deviation of the thickness of sheets produced on this shift.Data from in Exercise 9.41Plastic sheets produced by a machine are periodically monitored for possible fluctuations in thickness. Uncontrollable
Using the t able for the χ2 distributions, find:(a) The upper 5% point when d.f. = 8.(b) The upper 1% point when d.f. = 16.(c) The lower 2.5% point when d.f. = 9.(d) The lower 1% point when d.f. = 27.(e) Verify your answers above using software.
Use software to find the probability(a) χ2 ≤ 17.2 when d.f. = 9.(b) χ2 > 13.7 when d.f. = 23.(c) Find c such that P(χ2 > c) = .015 when d.f. = 12.(d) Find c such that P(χ2 < c) = .04 when d.f. = 33.
Referring to Exercise 9.21 , construct a 95% confidence interval for the population standard deviation of the diameters of Indian mounds.Data from in Exercise 9.21The following measurements of the diameters (in feet) of Indian mounds in southern Wisconsin were gathered by examining reports in
Do the data in Exercise 9.22 substantiate the conjecture that the true standard deviation of the acidity measurements is larger than 0.4? Test at a = .05.Data from in Exercise 9.22Measurements of the acidity (pH) of rain samples were recorded at 13 sites in an industrial region.Determine a 95%
Determine a 99% confidence interval for μ using the data in Exercise 9.49.Data from in Exercise 9.49Measurements of the amount of suspended solids in river water on 14 Monday mornings yield x̅ = 47 and s = 9.4 mg/ 1. Obtain a 95% confidence interval for the mean amount of suspended solids. State
Referring to Exercise 9 .55, test H0 : μ = 2.8 versus H1 : μ ≠ 2.8 using a = .02.Data from in Exercise 9 .55In a study of head injuries to ice hockey players the number of hits were monitored electronically by equipping approved helmets with accelerometers that measured head
Use software to find(a) χ2.05 with d.f. = 3 .(b) χ2.025 with d.f. = 24.(c) The lower .05 point with d.f. = 7.(d) The lower .025 point with d.f. = 32.
Use software to find(a) The 90th percentile of χ2 when d.f. = 13.(b) The 10th percentile of χ2 when d.f. = 4.(c) The median of χ2 when d.f. = 32.(d) The 1st percentile of χ2 when d.f. = 46.
Refer t o the data of Exercise 9.60. Is there strong evidence that the standard deviation for the efficiency of the new model is below 3.0?Data from in Exercise 9.60Combustion efficiency measurements were recorded for 10 home heating furnaces of a new model. The sample mean and standard deviation
A scientist developing a dietary supplement for rabbits decides to give a new supplement to IO rabbits and but not to 10 other rabbits who will serve as a control.(a) Briefly explain why it is important to randomly divide the rabbits in two groups. What might go wrong if the control group consists
Three male and three female students recorded the number of times they used a credit card in one week.(a) Calculate s2pooled.(b) Give an estimate of the common standard deviation for the number of uses.(c) Evaluate the t statistic for testing equality of the two population mean number of uses.(d)
One semester, an instructor taught the same computer course at two universities located in different cities. He was able to give the same final at both locations. The student's scores provided the summary statistics.(a) Obtain a point estimate of μ1 - μ2.(b) Estimate the S.E. (x̅ - Y̅).(c)
Grades for first semester are compared to those for second semester. The five one-semester courses biology, chemistry, English , history, and psychology must be taken next year.(a) Make a list of all possible ways to split the courses into two groups where the first group has two courses to be
According to the assumptions underlying the two independent samples design, let the first population be N (5, 3) and the second be N (9, 6). Identify the values of the parameters μ1, σ1, μ2, σ2.
Given the samples 4 6 5 and 9 6 3(a) Evaluate the d.f. for testing H0 : μ1 = μ2 = 0 using the approximate t statistic.(b) Use software to confirm your d.f. in part (a) .R: t.test(x,y) after x=c( 4,6,5) and y= c(9,6,3)
Given the following matched pairs sample,(a) Evaluate the t statistic(b) How many degrees of freedom does this t have? x T9 487 y 546232
Randomly allocate 2 subjects from among Al, Bob, Carol, Ellen, John to be in the control group. The others will receive a treatment.
Randomly allocate three subjects from among 6 mice, Alpha, Tau, Omega, Pi, Beta, Phi to group 1.
Two sun blockers are to b e compared. One blocker is rubbed on one side of a subject's back and the other blocker is rubbed on the other side. Each subject then lies in the sun for two hours. After waiting an additional hour, each side is rated according to redness.(a) Evaluate the t statistic(b)
An experiment was conducted to study whether cloud seeding reduces the occurrences of h ail. At a hail-prone geographical area, seeding was done on 50 stormy days and another 155 stormy days were also observed without seeding. The following counts were obtained .Do these data substantiate the
Two methods of memorizing difficult material are being tested to determine if one produces better retention. Nine pairs of students are included in the study. The students in each p air are matched according to IQ and academic background and then assigned to the two methods at random. A
A study (courtesy of R. Golubjatnikov) is undertaken to compare the rates of prevalence of CF antibody to parainfluenza I virus among boys and girls in the age group 5 to 9 years. Among 113 boys tested, 34 are found to have the antibody; among 139 girls tested, 54 have the antibody. Do the data
Consider the following linear regression modelwhere β0 = 4, β1 3, and the normal random variable e has the standard deviation 4.(a) What is the mean of the response Y when x = 4? When x = 5?(b) Will the response at x = 5 always be larger than that at x = 4? Explain. Y = B₁ + B₁x + e Bo
Identify the values of the parameters β0, β1, and σ in the statistical modelY = -3 + 4x + ewhere e is a normal random variable with mean 0 and standard deviation 2.
A help desk devoted to student software problems also receives phone calls. The number of persons that can be served in person, within one hour, is the response y . The predictor variable, x, is the number of phone calls answered.(a) Calculate x̅, y̅ , Sxx, Sxy, and SYY.(b) Calculate the least
Consider the linear regression modely = β0 + β1x + ewhere 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 larger than that at x = 5? Explain.
A student hourly employee does small secret arial projects.The number of projects she completes in a day is the response variable y . The number of hours she works in a day is the predictor variable x .(a) Calculate x̅, y̅, Sxx, Sxy, and SYY.(b) Calculate the least squares estimates β̂0 and
Using the formulas of β̂1 and SSE, show that SSE can also be expressed as(a)(b) SSE = S..... yy B₁ Sxv xy
An engineer found that by adding small amounts of a compound to rechargeable batteries during manufacture, she could extend their lifetimes. She experimented with different amounts of the additive ( g) and measured the hours they lasted in a laptop.(a) Calculate the least squares estimates β̂0
One measure of the development of a country is the Human Development Index (HDI). Life expectancy, literacy, educational attainment, and gross domestic product per capita are combined into an index between O and l , inclusive with l being the highest development. The United Nations Development
Referring to the formulas of β̂0 and β̂1 , show that the point (x̅, y̅) lies on the fitted regression line.
(a) Show that the sample correlation coefficient rand the slope Pi of the fitted regression line are related as(b) Show that SSE = (1 -r2) Syy· r || BI√Sxx XX IS..... yy
Based on a study of over 143,000 students, from 110 schools, investigators2 estimate the correlation between SAT scores and first year college grade point ratios by r = .35. If you predict first year college grades from the SAT score, what percentage of the variation in grades would be explained by
Many college students obtain college degree credits by demonstrating their proficiency on exams developed as part of the College Level Examination Program (CLEP). Based on their scores on the College Qualification Test (CQT), it is helpful if students could predict their scores on the corresponding
A recent graduate moving to a new job collected a sample of monthly rent (dollars) and size (square feet) of 2 bedroom apartments in one area of a midwest city.(a) Plot the scatter diagram and find the least squares fit of a straight line.(b) Do these data substantiate the claim that the monthly
Show that the SS due to regression, S2xy/ Sxx, can also be expressed as β̂21 Sxx·
A top Internet site features used cars for sale. The age, x, measured in years and the price, y, expressed in thousands of dollars were recorded from a sample of listings for a popular midsize foreign car.(a) Plot the scatter diagram.(b) Determine the equation of the least squares regression line
Refer to Exercise 11.51.Data from in Exercise 11.51A top Internet site features used cars for sale. The age, x, measured in years and the price, y, expressed in thousands of dollars were recorded from a sample of listings for a popular midsize foreign car.(a) From the fitted regression line,
Consider the data on a response variable and two predictor variables.(a) Fit a multiple regression model with response variable y.(b) Verify your answer using software. 지 X2 y -1 -1 -1 1 1 10 -1 11 -1 -1 1 1 1 1 -1 1 -1 5 14 1 12 1 5 16
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 Part a using software. X
Given the data,(a) Fit a quadratic curve using the equations on page 391.(b) Verify your answers using software.R: summary(lm(y ~ x + x2)) where x and y contain the observations and x2 = x*x. X У -2 5.0 -1 2.3 0 -.1 1 .7 2 2.1
Consider the data on a response variable and two predictor variables.(a) Fit a multiple regression model with response variable y.(b) Verify your answer using software.R: summary(lm( y ,.., xl + x2)) where xl, x2 contain the values of the predictor variables and y the responses. lx X2 y -1 -1 -1 -1

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