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Introduction To Mathematical Statistics And Its Applications 5th Edition Richard J. Larsen, Morris L. Marx - Solutions

An urn contains ten chips. An unknown number of the chips are white; the others are red. We wish to testH0: exactly half the chips are white versusH1: more than half the chips are whiteWe will draw, without replacement, three chips and reject H0 if two or more are white. Find α. Also, find β when
Suppose that a random sample of size 5 is drawn from a uniform pdf:We wish to testH0: θ =2versusH1: θ > 2by rejecting the null hypothesis if ymax ≥ k. Find the value of k that makes the probability of committing a Type I error equal to 0.05.
A sample of size 1 is taken from the pdffY(y) = (θ +1)yθ, 0 ≤ y ≤ 1The hypothesis H0: θ = 1 is to be rejected in favor of H1: θ > 1 if y ≥ 0.90.What is the test's level of significance?
A series of n Bernoulli trials is to be observed as data for testingH0: p = 1/2versusH1: p > 1/2The null hypothesis will be rejected if k, the observed number of successes, equals n. For what value of p will the probability of committing a Type II error equal 0.05?
Let X1 be a binomial random variable with n = 2 and pX1 = P(success). Let X2 be an independent binomial random variable with n = 4 and pX2 = P(success). Let X = X1 + X2. Calculate α ifH0: pX1 = pX2 = 1/2versusH1: pX1 = pX2 > ½is to be tested by rejecting the null hypothesis when k
A sample of size 1 from the pdf fY(y) = (1 + θ)yθ, 0 ≤ y ≤ 1, is to be the basis for testingH0: θ = 1versusH1: θ < 1The critical region will be the interval y ≤ 1/2. Find an expression for 1 − β as a function of θ.
An experimenter takes a sample of size 1 from the Poisson probability model, pX(k) = e−λλk/k!, k = 0, 1, 2, . . . , and wishes to testH0: λ = 6versusH1: λ < 6by rejecting H0 if k ≤ 2.(a) Calculate the probability of committing a Type I error.(b) Calculate the probability of committing a
A sample of size 1 is taken from the geometric probability model, pX(k) = (1 − p)k−1 p, k = 1, 2, 3, . . . , to test H0: p = 1/3 versus H1: p > 1/3 . The null hypothesis is to be rejected if k ≥ 4. What is the probability that a Type II error will be committed when p = 1/2?
Carry out the details to verify the decision rule change cited on p. 371 in connection with Figure 6.4.6.
Suppose that one observation from the exponential pdf, fY(y) = λe−λy, y > 0, is to be used to test H0: λ = 1 versus H1: λ < 1. The decision rule calls for the null hypothesis to be rejected if y ≥ ln 10. Find β as a function of λ.
A random sample of size 2 is drawn from a uniform pdf defined over the interval [0, θ]. We wish to testH0: θ = 2versusH1:θ < 2by rejecting H0 when y1 + y2 ≤ k. Find the value for k that gives a level of significance of 0.05.
Suppose that the hypotheses of Question 6.4.21 are to be tested with a decision rule of the form "Reject H0: θ = 2 if y1 y2 ≤ k∗." Find the value of k∗ that gives a level of significance of 0.05.
For the decision rule found in Question 6.2.2 to test H0: μ = 95 versus H1: μ ≠ 95 at the α = 0.06 level of significance, calculate 1 − β when μ = 90.
Construct a power curve for the α = 0.05 test of H0: μ = 60 versus H1: μ ≠ 60 if the data consist of a random sample of size 16 from a normal distribution having σ = 4.
If H0: μ = 240 is tested against H1: μ < 240 at the α = 0.01 level of significance with a random sample of twenty-five normally distributed observations, what proportion of the time will the procedure fail to recognize that μ has dropped to 220? Assume that σ = 50.
Suppose n = 36 observations are taken from a normal distribution where σ = 8.0 for the purpose of testing H0: μ = 60 versus H1: μ ≠ 60 at the α = 0.07 level of significance. The lead investigator skipped statistics class the day decision rules were being discussed and intend to reject H0 if y
If H0: μ = 200 is to be tested against H1: μ < 200 at the α = 0.10 level of significance based on a random sample of size n from a normal distribution where σ = 15.0, what is the smallest value for n that will make the power equal to at least 0.75 when μ=197?
Will n = 45 be a sufficiently large sample to test H0: μ = 10 versus H1: μ ≠ 10 at the α = 0.05 level of significance if the experimenter wants the Type II error probability to be no greater than 0.20 when μ = 12? Assume that σ = 4.
If H0: μ = 30 is tested against H1: μ > 30 using n = 16 observations (normally distributed) and if 1 − β = 0.85 when μ = 34, what does α equal? Assume that σ = 9.
Let k1, k2, . . . , kn be a random sample from the geometric probability functionpX(k; p) = (1 − p)k−1p, k = 1, 2, . . .Find λ, the generalized likelihood ratio for testing H0: p = p0 versus H1: p ≠ p0.
Let y1, y2, . . . , y10 be a random sample from an exponential pdf with unknown parameter λ. Find the form of the GLRT for H0: λ = λ0 versus H1: λ ≠ λ0.What integral would have to be evaluated to determine the critical value if α were equal to 0.05?
Let y1, y2, . . . , yn be a random sample from a normal pdf with unknown mean μ and variance 1. Find the form of the GLRT for H0: μ = μ0 versus H1: μ ≠ μ0.
In the scenario of Question 6.5.3, suppose the alternative hypothesis is H1: μ = μ1, for some particular value of μ1. How does the likelihood ratio test change in this case? In what way does the critical region depend on the particular value of μ1?
Let k denote the number of successes observed in a sequence of n independent Bernoulli trials, where p = P(success).(a) Show that the critical region of the likelihood ratio test of H0: p = 1/2 versus H1: p ≠ 1/2 can be written in the form k・ ln(k) + (n − k) ・ ln(n − k) ≥ λ∗∗(b)
Suppose a sufficient statistic exists for the parameter θ. Use Theorem 5.6.1 to show that the critical region of a likelihood ratio test will depend on the sufficient statistic.
Show directly—without appealing to the fact that χ2n is a gamma random variable—that fU (u) as stated in Definition 7.3.1 is a true probability density function.
Suppose that two independent samples of size n are drawn from a normal distribution with variance σ2. Let S21 and S22 denote the two sample variances. Use the fact that (n − 1)S2/σ2 has a chi square distribution with n −1 df to explain why
If the random variable F has an F distribution with m and n degrees of freedom, show that 1/F has an F distribution with n and m degrees of freedom.
Use the result claimed in Question 7.3.11 to express percentiles of fFn,m (r ) in terms of percentiles from fFm,n (r ). That is, if we know the values a and b for which P(a ≤ Fm,n ≤ b) = q, what values of c and d will satisfy the equation P(c ≤ Fn,m ≤ d) = q? “Check” your answer with
Show that as n→∞, the pdf of a Student t random variable with n df converges to fZ (z). (To show that the constant term in the pdf for Tn converges to 1/√2π, use Stirling’s formula, n! .= √2πn nne−n)Also, recall that (1 + a/n)n = ea.
Evaluate the integralusing the Student t distribution.
For a Student t random variable Y with n degrees of freedom and any positive integer k, show that E(Y2k) exists if 2k < n. (Integrals of the formare finite if α > 0, β > 0, and αβ > 1.)
Find the moment-generating function for a chi square random variable and use it to show that E(χ2n) = n and Var(χ2n) = 2n.
Is it believable that the numbers 65, 30, and 55 are a random sample of size 3 from a normal distribution with μ = 50 and σ = 10? Answer the question by using a chi square distribution. [Let Zi = (Yi − 50)/10 and use Theorem 7.3.1.]
Use the fact that (n − 1)S2/σ2 is a chi square random variable with n −1 df to prove thatVar(S2)= 2σ4/n −1
Let Y1, Y2, . . . , Yn be a random sample from a normal distribution. Use the statement of Question 7.3.4 to prove that S2 is consistent for σ2.Var(S2)= 2σ4/n −1
If Y is a chi square random variable with n degrees of freedom, the pdf of (Y − n)/√2n converges to fZ (z) as n goes to infinity (recall Question 7.3.2). Use the asymptotic normality of (Y − n)/√2n to approximate the fortieth percentile of a chi square random variable with 200 degrees of
Let V and U be independent chi square random variables with 7 and 9 degrees of freedom, respectively. Is it more likely that V/7/U/9 will be between (1) 2.51 and 3.29 or (2) 3.29 and 4.20?
Use Appendix Table A.4 to find the values of x that satisfy the following equations:(a) P(0.109 < F4,6 < x) = 0.95(b) P(0.427 < F11,7 < 1.69) = x(c) P(Fx,x > 5.35) = 0.01(d) P(0.115 < F3,x < 3.29) = 0.90(e) P (x < V/2/U/3) = 0.25, where V is a chi square random variable with
How long does it take to fly from Atlanta to New York’s LaGuardia airport? There are many components of the time elapsed, but one of the more stable measurements is the actual in-air time. For a sample of sixty-one flights between these destinations on Sundays in April, the time in minutes (y)
In a nongeriatric population, platelet counts ranging from 140 to 440 (thousands per mm3 of blood) are considered €œnormal.€ The following are the platelet counts recorded for twenty four female nursing home residents (169).Use the following sums:How does the definition of €œnormal€
If a normally distributed sample of size n=16 produces a 95% confidence interval for μ that ranges from 44.7 to 49.9, what are the values of y and s?
Two samples, each of size n, are taken from a normal distribution with unknown mean μ and unknown standard deviation σ. A 90% confidence interval for μ is constructed with the first sample, and a 95% confidence interval for μ is constructed with the second. Will the 95% confidence interval
Revenues reported last week from nine boutiques franchised by an international clothier averaged $59,540 with a standard deviation of $6860. Based on those figures, in what range might the company expect to find the average revenue of all of its boutiques?
The weather station at Dismal Swamp, California, recorded monthly precipitation (y) for twenty-eight years. For these data, = 1392.6 and = 10, 518.84.(a) Find the 95% confidence interval for the mean monthly precipitation.(b) The table on the right gives a frequency distribution for the Dismal
Recall the Bacillus subtilis data in Question 5.3.2. Test the null hypothesis that exposure to the enzyme does not affect a worker’s respiratory capacity (as measured by the FEV1/VC ratio). Use a one-sided H1 and let α = 0.05. Assume that σ is not known.
Assess the credibility of the theory that Etruscans were native Italians by testing an appropriate H0 against a two-sided H1. Set α equal to 0.05. Use 143.8 mm and 6.0 mm for y̅ and s, respectively, and let μo = 132.4. Do these data appear to satisfy the distribution assumption made by the t
MBAs R Us advertises that its program increases a person’s score on the GMAT by an average of forty points. As a way of checking the validity of that claim, a consumer watchdog group hired fifteen students to take both the review course and the GMAT. Prior to starting the course, the fifteen
In addition to the Shoshoni data of Case Study 7.4.2, a set of rectangles that might tend to the golden ratio are national flags. The table below gives the width-to-length ratios for a random sample of the flags of thirty-four countries. Let μ be the width-to-length ratio for national flags. At
A manufacturer of pipe for laying underground electrical cables is concerned about the pipe’s rate of corrosion and whether a special coating may retard that rate. As a way of measuring corrosion, the manufacturer examines a short length of pipe and records the depth of the maximum pit. The
Explain why the distribution of t ratios calculated from small samples drawn from the exponential pdf, fY(y) = e−y, y ≥ 0, will be skewed to the left [recall Figure 7.4.6(b)]. [What does the shape of fY(y) imply about the possibility of each yi being close to 0? If the entire sample did consist
Suppose one hundred samples of size n = 3 are taken from each of the pdfs (1) fY(y) = 2y, 0 ≤ y ≤ 1 and (2) fY(y) = 4y3, 0 ≤ y ≤ 1 and for each set of three observations, the ratio − μ/s/√3 is calculated, where μ is the expected value of the particular pdf being sampled. How would
On which of the following sets of data would you be reluctant to do a t test? Explain.(a)(b)(c)
Which of the following differences is larger? Explain.t.05,n − t.10,n or t.10,n − t.15,n
A random sample of size n =9 is drawn from a normal distribution with μ = 27.6. Within what interval (−a, +a) can we expect to find − 27.6/S/√9 80% of the time? 90% of the time?
Suppose a random sample of size n = 11 is drawn from a normal distribution with μ = 15.0. For what value of k is the following true?P(|− 15.0/S/√11| ≥ k) = 0.05
Let Y and S denote the sample mean and sample standard deviation, respectively, based on a set of n = 20 measurements taken from a normal distribution with μ = 90.6. Find the function k(S) for whichP[90.6 − k(S) ≤ Y̅ ≤ 90.6 + k(S)] = 0.99
Cell phones emit radio frequency energy that is absorbed by the body when the phone is next to the ear and may be harmful. The table in the next column gives the absorption rate for a random sample of twenty cell phones. (The Federal Communication Commission sets a maximum of 1.6 watts per kilogram
The following table lists the typical cost of repairing the bumper of a moderately priced midsize car damaged by a corner collision at 3 mph. Use these observations to construct a 95% confidence interval for μ, the true average repair cost for all such automobiles with similar damage. The sample
Creativity, as any numbers of studies have shown, is very much a province of the young. Whether the focus is music, literature, science, or mathematics, an individual’s best work seldom occurs late in life. Einstein, for example, made his most profound discoveries at the age of twenty-six;
How much interest certificates of deposit (CDs) pay varies by financial institution and also by length of the investment? A large sample of national one-year CD offerings in 2009 showed an average interest rate of 1.84 and a standard deviation σ = 0.262. A five-year CD ties up an investor’s
(a) Use the asymptotic normality of chi square random variables (see Question 7.3.6) to derive large-sample confidence interval formulas for σ and σ2.(b) Use your answer to part (a) to construct an approximate 95% confidence interval for the standard deviation of estimated potassium-argon ages
If a 90% confidence interval for σ2 is reported to be (51.47, 261.90), what is the value of the sample standard deviation?
Let Y1, Y2, . . . , Yn be a random sample of size n from the pdffY(y) = (1/θ)e−y/θ, y > 0; θ > 0(a) Use moment-generating functions to show that the ratio 2nY̅/θ has a chi square distribution with 2n df.(b) Use the result in part (a) to derive a 100(1 − α)% confidence interval for
Another method for dating rocks was used before the advent of the potassium-argon method described in Case Study 7.5.1. Because of a mineral’s lead content, it was capable of yielding estimates for this same time period with a standard deviation of 30.4 million years. The potassium-argon method
When working properly, the amounts of cement that a filling machine puts into 25-kg bags have a standard deviation (σ) of 1.0 kg. In the next column are the weights recorded for thirty bags selected at random from a day’s production. Test H0: σ2 = 1 versus H1: σ2 >1 using the α =0.05 level
A stock analyst claims to have devised a mathematical technique for selecting high-quality mutual funds and promises that a client’s portfolio will have higher average ten-year annualized returns and lower volatility; that is, a smaller standard deviation. After ten years, one of the analyst’s
For df values beyond the range of Appendix Table A.3, chi square cutoffs can be approximated by using a formula based on cutoffs from the standard normal pdf, fZ(z). Define χ2p,n and z∗p so that P (χ2n ≤ χ2p,n) = p and P(Z ≤ z∗p) = p, respectively. Thenχ2p,n = n(1− 2/9n
Let Y1, Y2, . . . , Yn be a random sample of size n from a normal distribution having mean μ and variance σ2. What is the smallest value of n for which the following is true?P(S2/σ2 < 2) ≥ 0.95
Start with the fact that (n − 1)S2/σ2 has a chi square distribution with n − 1 df (if the Yi ’s are normally distributed) and derive the confidence interval formulas given in Theorem 7.5.1.
A random sample of size n=19 is drawn from a normal distribution for which σ2 =12.0. In what range are we likely to find the sample variance, s2? Answer the question by finding two numbers a and b such thatP(a ≤ S2 ≤ b) = 0.95
How long sporting events last is quite variable. This variability can cause problems for TV broadcasters, since the amount of commercials and commentator blather varies with the length of the event. As an example of this variability, the table below gives the lengths for a random sample of
Compute −2 ln λ (see Equation 9.4.1) for the nightmare data of Case Study 9.4.2, and use it to test the hypothesis that pX = pY. Let α = 0.01.
In a study designed to see whether a controlled diet could retard the process of arteriosclerosis, a total of 846 randomly chosen persons were followed over an eight year period. Half were instructed to eat only certain foods; the other half could eat whatever they wanted. At the end of eight
Water witching, the practice of using the movements of a forked twig to locate underground water (or minerals), dates back over 400 years. Its first detailed description appears in Agricola’s De re Metallica, published in 1556. That water witching works remains a belief widely held among rural
If flying saucers are a genuine phenomenon, it would follow that the nature of sightings (that is, their physical characteristics) would be similar in different parts of the world. A prominent UFO investigator compiled a listing of 91 sightings reported in Spain and 1117 reported elsewhere. Among
In some criminal cases, the judge and the defendant’s lawyer will enter into a plea bargain, where the accused pleads guilty to a lesser charge. The proportion of time this happens is called the mitigation rate. A Florida Corrections Department study showed that Escambia County had the state’s
Suppose H0: pX = pY is being tested against H1: pX ≠ pY on the basis of two independent sets of one hundred Bernoulli trials. If x, the number of successes in the first set, is sixty and y, the number of successes in the second set, is forty-eight, what P-value would be associated with the data?
A total of 8605 students are enrolled full-time at State University this semester, 4134 of whom are women. Of the 6001 students who live on campus, 2915 are women. Can it be argued that the difference in the proportion of men and women living on campus is statistically significant? Carry out an
The kittiwake is a seagull whose mating behavior is basically monogamous. Normally, the birds separate for several months after the completion of one breeding season and reunite at the beginning of the next. Whether or not the birds actually do reunite, though, may be affected by the success of
A utility infielder for a National League club batted .260 last seasons in three hundred trips to the plate. This year he hit .250 in two hundred at-bats. The owners are trying to cut his pay for next year on the grounds that his output has deteriorated. The player argues, though, that his
In 1965 a silver shortage in the United States prompted Congress to authorize the minting of silver less dimes and quarters. They also recommended that the silver content of half-dollars be reduced from 90% to 40%. Historically, fluctuations in the amount of rare metals found in coins are not
Construct an 80% confidence interval for the difference pM − pW in the nightmare frequency data summarized in Case Study 9.4.2.
If pX and pY denote the true success probabilities associated with two sets of n and m independent Bernoulli trials, respectively, the ratiohas approximately a standard normal distribution. Use that fact to prove Theorem 9.5.3.
Suicide rates in the United States tend to be much higher for men than for women, at all ages. That pattern may not extend to all professions, though. Death certificates obtained for the 3637 members of the American Chemical Society who died over a twenty-year period revealed that 106 of the 3522
Male fiddler crabs solicit attention from the opposite sex by standing in front of their burrows and waving their claws at the females who walk by. If a female likes what she sees, she pays the male a brief visit in his burrow. If everything goes well and the crustacean chemistry clicks, she will
Construct two 99% confidence intervals for μX − μY using the data of Case Study 9.2.3, first assuming the variances are equal, and then assuming they are not.
Carry out the details to complete the proof of Theorem 9.5.1.
Suppose that X1, X2, . . . , Xn and Y1, Y2, . . . , Ym are independent random samples from normal distributions with means μX and μY and known standard deviations σX and σY, respectively. Derive a 100(1−α)% confidence interval for μX − μY.
Construct a 95% confidence interval for σ2X/σ2Y based on the data in Case Study 9.2.1. The hypothesis test referred to tacitly assumed that the variances were equal. Does that agree with your confidence interval? Explain.
One of the parameters used in evaluating myocardial function is the end diastolic volume (EDV). The following table shows EDVs recorded for eight persons considered to have normal cardiac function and for six with constrictive pericarditis (192). Would it be correct to use Theorem 9.2.2 to test H0:
Complete the proof of Theorem 9.5.2.
Flonase is a nasal spray for diminishing nasal allergic symptoms. In clinical trials for side effects, 782 sufferers from allergic rhinitis were given a daily dose of 200 mcg of Flonase. Of this group, 126 reported headaches. A group of 758 subjects were given a placebo, and 111 of them reported
Some states that operate a lottery believe that restricting the use of lottery profits to supporting education makes the lottery more profitable. Other states permit general use of the lottery income. The profitability of the lottery for a group of states in each category is given below.Test at the
A company markets two brands of latex paint—regular and a more expensive brand that claims to dry an hour faster. A consumer magazine decides to test this claim by painting ten panels with each product. The average drying time of the regular brand is 2.1 hours with a sample standard deviation of
(a) Suppose H0: μX = μY is to be tested against H1: μX ≠ μY. The two sample sizes are 6 and 11. If sp = 15.3, what is the smallest value for | x̅ − y̅| that will result in H0 being rejected at the α =0.01 level of significance?(b) What is the smallest value for x̅ − y̅ that
Suppose that H0: μX = μY is being tested against H1: μX ≠ μY, where σ2X and σ2Y are known to be 17.6 and 22.9, respectively. If n =10, m =20, x̅ = 81.6, and y̅ = 79.9, what P-value would be associated with the observed Z ratio?
An executive has two routes that she can take to and from work each day. The first is by interstate; the second requires driving through town. On the average it takes her 33 minutes to get to work by the interstate and 35 minutes by going through town. The standard deviations for the two routes are
Prove that the Z ratio given in Equation 9.2.1 has a standard normal distribution.
If X1, X2, . . . , Xn and Y1, Y2, . . . , Ym are independent random samples from normal distributions with the same σ2, prove that their pooled sample variance, s2p, is an unbiased estimator for σ2.

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