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

The Master of Business Administration (M.B.A.) degree typically prepares its possessors for a high-salaried position, most often in business or industry. So, a reasonable measure of the effectiveness of an M.B.A. program is the median salary of its graduates five years after graduation. The table
The best straight line through the Massachusetts funding/graduation rate data described in Question 11.2.7 has the equation y =81.088+0.412x, where s = 11.78848.(a) Construct a 95% confidence interval for β1.(b) What does your answer to part (a) imply about the outcome of testing H0: β1 = 0
In the radioactive exposure example in Question 11.2.9, find the 95% confidence interval for E(Y |9.00) and the prediction interval for the value 9.00.
Attorneys representing a group of male buyers employed by Flirty Fashions are filing a reverse discrimination suit against the female-owned company. Central to their case are the following data, showing the relationship between years of service and annual salary for the firm€™s fourteen buyers,
Polls taken during a city€™s last two administrations (one Democratic, one Republican) suggested that public support of the two mayors fell off linearly with years in office. Can it be concluded from the following data that the rates at which the two administrations lost favor were
Prove that the variance of YË† can also be written
Show thatfor any set of points (xi , Yi), i = 1, 2, . . . , n.
Based on the data in Question 11.2.1, the relationship between y, the ambient temperature, and x, the frequency of a cricket’s chirping, is given by y = 25.2 + 3.29x, where s =3.83. At the α = 0.01 level of significance, can the hypothesis that chirping frequency is not related to temperature be
Suppose an experimenter intends to do a regression analysis by taking a total of 2n data points, where the xi’s are restricted to the interval [0, 5]. If the xy-relationship is assumed to be linear and if the objective is to estimate the slope with the greatest possible precision, what values
Suppose a total of n = 9 measurements are to be taken on a simple linear model, where the xi’s will be set equal to 1, 2, . . . , and 9. If the variance associated with the xy-relationship is known to be 45.0, what is the probability that the estimated slope will be within 1.5 units of the true
Prove the useful computing formula (Equation 11.3.5) that
The sodium nitrate (NaNO3) solubility data in Question 11.2.3 is described nicely by the regression line y = 67.508 + 0.871x, where s = 0.959. Construct a 90% confidence interval for the y-intercept, β0.
Set up and carry out an appropriate hypothesis test for the Hanford radioactive contamination data given in Question 11.2.9. Let α = 0.05. Justify your choice of H0 and H1. What do you conclude?
Test H0: β1 = 0 versus H1: β1 = 0 for the plumage index/behavioral index data given in Question 11.2.11 Let α = 0.05. Use the fact that y = 0.61 + 0.84x is the best straight line describing the xy-relationship.
Let X and Y have the joint pdfFind Cov(X, Y ) and Ï(X, Y ).
In Case Study 11.3.1, how much of the variability in CHD mortality is explained by cigarette consumption?
Some baseball fans believe that the number of home runs a team hits is markedly affected by the altitude of the club€™s home park. The rationale is that the air is thinner at the higher altitudes, and balls would be expected to travel farther. The following table shows the altitudes (X) of
The extent to which stress is a contributing factor to the severity of chronic illnesses was the focus of the study summarized in the following table (208). Seventeen conditions were compared on a Seriousness of Illness Rating Scale (SIRS). Patients with each of these conditions were asked to fill
Among the many strategies that investors use to try to predict trends in the stock market is the €œearly warning€ system, which is based on the premise that what the market does in the first week in January is indicative of what it will do over the next twelve months. Listed in the following
Suppose that X and Y have the joint pdffX,Y(x, y) = x + y, 0 < x < 1, 0 < y < 1Find ρ(X, Y).
If the random variables X and Y have the joint pdfshow that Cov(X, Y) = 8/450. Calculate Ï(X, Y).
Suppose that X and Y are discrete random variables with the joint pdf(x, y) fX,Y(x, y)(1, 2)………………………..1/2(1, 3)..………………………1/4(2, 1)..………………………1/8(2, 4)..………………………1/8Find the correlation coefficient between X and Y.
Prove that ρ(a + bX, c + dY) = ρ(X, Y) for constants a, b, c, and d where b and d are positive. This result allows for a change of scale to one convenient for computation.
Let the random variable X take on the values 1, 2, . . . , n, each with probability 1/n. Define Y to be X2. Find ρ(X, Y) and ρ(X, Y ).
(a) For random variables X and Y , show thatCov(X + Y, X − Y ) = Var(X) − Var(Y)(b) Suppose that Cov(X, Y) = 0. Prove thatρ(X + Y, X − Y ) = Var(X) − Var(Y)/Var(X) + Var(Y)
Derive Equation 11.4.2 from Equation 11.4.1.
Let (x1, y1), (x2, y2), . . . , (xn, yn) be a set of measurements whose sample correlation coefficient is r . Show thatwhere 1 is the maximum likelihood estimate for the slope.
Suppose that X and Y have a bivariate normal pdf with μX = 3, μY = 6, σ2X = 4, σ2Y = 10, and ρ = 1/2. Find P(5 < Y < 6 1/2 ) and P(5 < Y < 6 1/2| x = 2).
Test H0: ρ = 0 versus H1: ρ = 0 for the SRE/SIRS data described in Question 11.4.13. Let 0.01 be the level of significance.
The National Collegiate Athletic Association has had a long-standing concern about the graduation rate of athletes. Under the urging of the Association, some prominent athletic programs increased the funds for tutoring athletes. The table below gives the amount spent (in millions of dollars) and
Suppose that X and Y have a bivariate normal distribution with Var(X) = Var(Y).(a) Show that X and Y − ρX are independent.(b) Show that X + Y and X − Y are independent.
Suppose that X and Y have a bivariate normal distribution.(a) Prove that X +Y has a normal distribution when X and Y are standard normal random variables.(b) Find E(cX + dY) and Var(cX + dY) in terms of μX,μY,σX,σY , and ρ(X, Y), where X and Y are arbitrary normal random variables.
Suppose that the random variables X and Y have a bivariate normal pdf with μX = 56, μY =11,σ2X = 1.2,σ2Y = 2.6, and ρ = 0.6. Compute P(10 < Y < 10.5 | x = 55). Suppose that n = 4 values were to be observed with x fixed at 55. Find P(10.5 < < 11 | x = 55).
Give conditions on a > 0, b > 0, and u so thatfX,Y(x, y) = ke−(ax2−2uxy+by2)is the bivariate normal density of random variables X and Y each having expected value 0. Also, find Var(X), Var(Y), and ρ(X, Y).
What would the conclusion be for the test of Example 11.5.1 if α =0.01?
In a study of heart disease (73), the weight (in pounds) and the blood cholesterol (in mg/dl) of fourteen men without a history of coronary incidents were recorded. At the Î± = 0.05 level, can we conclude from these data that the two variables are independent?The data in the table give the
Recall the baseball data in Question 11.4.11. Test whether home run frequency and home park altitude are independent. Let α = 0.05.
The following are the gas mileages recorded during a series of road tests with four new models of Japanese luxury sedans. Test the null hypothesis that all four models, on the average, give the same mileage. Let Î± = 0.05. Will the conclusion change if Î± = 0.10?
Use Fisher€™s Lemma to prove Theorem 12.2.2.
Verify the conclusion of Example 12.2.2 by doing a t test and an analysis of variance on the data of Question 9.2.8. Show that the observed F ratio is the square of the observed t ratio and that the F critical value is the square of the t critical value.
Do an analysis of variance on the Mark Twain–Quintus Curtius Snodgrass data of Case Study 9.2.1. Verify that the observed F ratio is the square of the observed t ratio.
Do an analysis of variance and a pooled two sample t test on the motorcycle data given in Question 8.2.2. How are the observed F ratio and observed t ratio related? How are the two critical values related? Assume that α = 0.05.
Mount Etna erupted in 1669, 1780, and 1865. When molten lava hardens, it retains the direction of the Earth€™s magnetic field. Three blocks of lava were examined from each of these eruptions and the declination of the magnetic field in the block was measured (170). The results are given in the
An indicator of the value of a stock relative to its earnings is its price-earnings ratio: the average of a given year€™s high and low selling prices divided by its annual earnings. The following table provides the price-earnings ratios for a sample of thirty stocks, ten each from the
Each of five varieties of corn are planted in three plots in a large field. The respective yields, in bushels per acre, are in the following table.Test whether the differences among the average yields are statistically significant. Show the ANOVA table. Let 0.05 be the level of significance.
Three pottery shards from four widely scattered and now-extinct Native American tribes have been collected by a museum. Archaeologists were asked to estimate the age of the shards. Based on the results shown in the following table, is it conceivable that the four tribes were contemporaries of one
Recall the teachers’ expectation data described in Question 8.2.7. Let μj denote the true average IQ change associated with group j, j = I, II, or III. Test H0: μI = μII = μIII versus H1: not all μj’s are equal. Let α = 0.05.
Fill in the entries missing from the following ANOVA table.
Do the following data appear to violate the assumptions underlying the analysis of variance? Explain.
Prove Equations 12.2.4 and 12.2.5.
Use Tukey’s method to make all the pairwise comparisons for the heart rate data of Case Study 12.2.1 at the 0.05 level of significance.
Construct 95% Tukey intervals for the three pairwise differences, Î¼i ˆ’ Î¼j, for the data of Question 12.2.3.
Intravenous infusion fluids produced by three different pharmaceutical companies (Cutter, Abbott, and McGaw) were tested for their concentrations of particulate contaminants. Six samples were inspected from each company. The figures listed in the table are, for each sample, the number of particles
Construct 95% Tukey intervals for all ten pairwise differences, μi − μj, for the data of Question 12.2.4. Summarize the results by plotting the five sample averages on a horizontal axis and drawing straight lines under varieties whose average yields are not significantly different.
Construct 95% Tukey confidence intervals for the three pairwise differences associated with the murder culpability scores described in Question 8.2.15.Which differences are statistically significant?
If 95% Tukey confidence intervals tell us to reject H0: μ1 = μ2 and H0: μ1 = μ3, will we necessarily reject H0: μ2 = μ3?
The cathode warm-up time (in seconds) was determined for three different types of X-ray tubes using fifteen observations of each type. The results are listed in the following table.Do an analysis of variance on these data and test the hypothesis that the three tube types require the same average
Test the hypothesis that the average of the true yields for the first three varieties of corn described in Question 12.2.4 is the same as the average for the last two. Let α = 0.05.
In Case Study 12.2.1 test the hypothesis that the average of the heart rates for light and moderate smokers is the same as that for heavy smokers. Let the level of significance be 0.05.
Large companies have the option of limiting their growth, but does doing so lead to higher profitability? The table below gives the profitability for a sample of twenty-one top-ranked companies, where profitability is expressed in terms of annual profit as a percentage of total company assets. The
Verify that C3 = 11/12μA + 11/12μB − μC – 5/6μD is orthogonal to the C1 and C2 of Case Study 12.4.1. Find SSC3 and illustrate the statement of Theorem 12.4.1.
For many years sodium nitrite has been used as a curing agent for bacon, and until recently it was thought to be perfectly harmless. But now it appears that during frying, sodium nitrite induces the formation of nitrosopyrrolidine (NPy), a substance suspected of being a carcinogen. In one study
A commercial film processor is experimenting with two kinds of fully automatic color developers. Six sheets of exposed film are put through each developer. The number of flaws on each negative visible with the naked eye is then counted.Number of Visible FlawsDeveloper A Developer
An experimenter wants to do an analysis of variance on a set of data involving five treatment groups, each with three replicates. She has computed .j and Sj for each group and gotten the results listed in the following tableWhat should the experimenter do before computing the various sums of
Three air-to-surface missile launchers are tested for their accuracy. The same gun crew fires four rounds with each launcher, each round consisting of twenty missiles. A €œhit€ is scored if the missile lands within ten yards of the target. The following table gives the number of hits
In the scenario of the previous question, is H1: μ1 = 2, μ2 = 1, μ3 = 1, μ4 = −3, μ5 = 0 an “admissible” alternative hypothesis?
If the random variable V has a noncentral χ2 distribution with r degrees of freedom and noncentrality parameter γ, use its moment-generating function to find E(V).
Suppose V1, V2, . . . , Vn are independent noncentral χ2 random variables having r1, r2, . . . , rn degrees of freedom, respectively, and with noncentrality parameters γ1,γ2, . . . , γn. Find the distribution of V = V1 + V2 +· · ·+ Vn.
In recent years a number of research projects in extrasensory perception have examined the possibility that hypnosis may be helpful in bringing out ESP in persons who did not think they had any. The obvious way to test such a hypothesis is with a self-paired design: the ESP ability of a subject
Refer to the rat poison data of Case Study 13.2.2. Partition the treatment sum of squares into three orthogonal contrasts. Let one contrast test the hypothesis that the true acceptance percentage for the plain cornmeal is equal to the true acceptance percentage for the cornmeal with artificial
Prove the computing formulas given in Equations 13.2.2, 13.2.3, and 13.2.4.
Differentiate the function
True or false:(a)(b) Either MSTR or MSB or both are greater than or equal to MSE.
For a set of randomized block data comparing k treatments within b blocks, find(a) E(SSB)(b) E(SSE)
The following table shows the audience shares of the three major networks€™ evening news broadcasts in four major cities as reported by Arbitron. Test at the Î± = 0.10 level of significance the null hypothesis that viewing levels for news are the same for ABC, CBS, and NBC.
A paint manufacturer is experimenting with an additive that might make the paint less chalky. To ensure that the additive does not affect the tint, a quality-control engineer takes a sample from each of seven batches of Osage Orange.Each sample is split in half, and the additive is put into one of
The number of new building permits can be a good indicator of the strength of a region€™s economic growth. The following table gives percentage increases over a four-year period for three geographical areas. Analyze the data. Let Î± = 0.05. What are your conclusions?
nalyze the Transylvania effect data in Case Study 13.2.3 by calculating 95% Tukey confidence intervals for the pairwise differences among the admission rates for the three different phases of the moon. How do your conclusions agree with (or differ from) those already discussed on p. 640? Let
The table below gives a stock fund€™s quarterly returns for the years 2003 to 2007. Are the results affected by the quarter of the year? Is the variability in the return from year to year statistically significant? State your conclusions using the Î± = 0.05 level of significance.
Find the 95% Tukey intervals for the data of Question 13.2.2, and use them to test the three pairwise comparisons of ABC, CBS, and NBC.
A comparison was made of the efficiency of four different unit-dose injection systems. A group of pharmacists and nurses were the €œblocks.€ For each system, they were to remove the unit from its outer package, assemble it, and simulate an injection. In addition to the standard system of
Heart rates were monitored (10) for six tree shrews (Tupaia glis) during three different stages of sleep: LSWS (light slow-wave sleep), DSWS (deep slow-wave sleep), and REM (rapid-eye-movement sleep).(a) Do the analysis of variance to test the equality of the heart rates during these three phases
Case Study 7.5.2 compared the volatility of Global Rock Funds€™ return on investments to that of the benchmark Lipper fund. But can it be said that the returns themselves beat the benchmark? The table below gives the annual returns of the Global Rock Fund for the years 1989 to 2007 and the
Recall the depth perception data described in Question 8.2.6. Use a paired t test with α = 0.05 to compare the numbers of trials needed to learn depth perception for Mothered and Unmothered lambs.
Blood coagulates as a result of a complex sequence of chemical reactions. The protein thrombin triggers the clotting of blood under the influence of another protein called prothrombin. One measure of a person's blood clotting ability is expressed in prothrombin time, which is defined to be the
Use a paired t test to analyze the hypnosis/ESP data given in Question 13.2.1. Let α = 0.05.
Perform the hypothesis test indicated in Question 13.2.3 at the 0.05 level using a paired t test. Compare the square of the observed t with the observed F. Do the same for the critical values associated with the two procedures. What would you conclude?
Let D1, D2, . . . , Db be the within-block differences as defined in this section. Assume that the Di 's are normal with mean μD and variance σ2D , for i = 1, 2, . . . , b. Derive a formula for a 100(1 − α)% confidence interval for μD. Apply this formula to the data of Case Study 13.3.1 and
Construct a 95% confidence interval for μD in the prothrombin time data described in Question 13.3.3. See Question 13.3.6.
Show that the paired t test is equivalent to the F test in a randomized block design when the number of treatment levels is two. (Consider the distribution of T2 = b2/S2D.)
Recall the data in Question 8.2.9 giving the sizes of 10 gorilla groups studied in the Congo. Is it believable that the true median size, ˜μ, of all such groups is 9? Answer the question by finding the P-value associated with the null hypothesis H0: ˜μ = 9. Assume that H1 is two-sided.
Suppose that a random sample of size 36, Y1, Y2, . . . , Y36, is drawn from a uniform pdf defined over the interval (0, θ), where θ is unknown. Set up a large sample sign test for deciding whether or not the 25th percentile of the Y-distribution is equal to 6. Let α = 0.05. With what probability
Use a small-sample sign test to analyze the aerobics data given in Case Study 13.3.1. Use the binominal distribution displayed in Question 14.2.1. Let α = 0.05. Does your conclusion agree with the inference drawn from the paired t test?
Test H0: ˜μ = 0.12 versus H1: ˜μ < 0.12 for the release chirp data given in Question 8.2.12. Compare the P-value associated with the large-sample test described in Theorem 14.2.1 with the exact P-value based on the binomial distribution.
Below are n = 50 observations generated by Minitab€™s RANDOM command that are presumably a random sample from the exponential pdf, fY(y) = eˆ’y, y ‰¥ 0. Use Theorem 14.2.1 to test whether the difference between the sample median for these yi€™s (= 0.604) and the true median of fY(y)
Let Y1, Y2, . . . , Y22 be a random sample of normally distributed random variables with an unknown mean μ and a known variance of 6.0.We wish to testH0: μ = 10versusH1: μ > 10Construct a large-sample sign test having a Type I error probability of 0.05. What will the power of the test be if
Suppose that n = 7 paired observations, (Xi, Yi), are recorded, i = 1, 2, . . . , 7. Let p = P(Yi > Xi). Write out the entire probability distribution for Y+, the number of positive differences among the set of Yi − Xi’s, i = 1, 2, . . . , 7, assuming that p = 1/2. What α levels are
Analyze the Shoshoni rectangle data (Case Study 7.4.2) with a sign test. Let α = 0.05.
Recall the FEV1/VC data described in Question 5.3.2. Test H0: ˜μ = 0.80 versus H0: ˜μ < 0.80 using a sign test. Compare this conclusion with that of a t test of H0: μ = 0.80 versus H1: μ < 0.80. Let α = 0.10. Assume that σ is unknown.
Do a sign test on the ESP data in Question 13.2.1. Define H1 to be one-sided, and let α = 0.05.

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