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Statistics
For the Major League Baseball seasons from 1950 through 2008, there were fifty-nine nine-inning games in which one of the teams did not manage to get a hit. The data in the table give the number of
A commonly used IQ test is scaled to have a mean of 100 and a standard deviation of σ = 15. A school counselor was curious about the average IQ of the students in her school and took a random sample
In 1927, the year he hit sixty home runs, Babe Ruth batted .356, having collected 192 hits in 540 official at-bats (140). Based on his performance that season, construct a 95% confidence interval for
To buy a thirty-second commercial break during the telecast of Super Bowl XXIX cost approximately $1,000,000. Not surprisingly, potential sponsors wanted to know how many people might be watching. In
The Pew Research Center did a survey of 2253 adults and discovered that 63% of them had broadband Internet connections in their homes. The survey report noted that this figure represented a
If (0.57, 0.63) is a 50% confidence interval for p, what does k/n equal and how many observations were taken?
Suppose a coin is to be tossed n times for the purpose of estimating p, where p = P(heads). How large must n be to guarantee that the length of the 99% confidence interval for p will be less than
On the morning of November 9, 1994-the day after the electoral landslide that had returned Republicans to power in both branches of Congress-several key races were still in doubt. The most prominent
Examine the first two derivatives of the function g(p) = p(1 − p) to verify the claim on p. 305 that p(1 − p) ≤ 1/4 for 0 < p < 1.
The financial crisis of 2008 highlighted the issue of excessive compensation for business CEOs. In a Gallup poll in the summer of 2009, 998 adults were asked, "Do you favor or oppose the federal
The production of a nationally marketed detergent results in certain workers receiving prolonged exposures to a Bacillus subtilis enzyme. Nineteen workers were tested to determine the effects of
Viral infections contracted early during a woman's pregnancy can be very harmful to the fetus. One study found a total of 86 deaths and birth defects among 202 pregnancies complicated by a
Rewrite Definition 5.3.1 to cover the case where a finite correction factor needs to be included (i.e., situations where the sample size n is not negligible relative to the population size N).
A public health official is planning for the supply of influenza vaccine needed for the upcoming flu season.She took a poll of 350 local citizens and found that only 126 said they would be
Given that n observations will produce a binomial parameter estimator, X/n, having a margin of error equal to 0.06, how many observations are required for the proportion to have a margin of error
Given that a political poll shows that 52% of the sample favors Candidate A, whereas 48% would vote for Candidate B, and given that the margin of error associated with the survey is 0.05, does it
Assume that the binomial parameter p is to be estimated with the function X n , where X is the number of successes in n independent trials. Which demands the larger sample size: requiring that X/n
Suppose that p is to be estimated by X/n and we are willing to assume that the true p will not be greater than 0.4. What is the smallest n for which X/n will have a 99% probability of being within
Let p denote the true proportion of college students who support the movement to colorize classic films. Let the random variable X denote the number of students (out of n) who prefer colorized
University officials are planning to audit 1586 new appointments to estimate the proportion p who have been incorrectly processed by the payroll department.(a) How large does the sample size need to
Mercury pollution is widely recognized as a serious ecological problem. Much of the mercury released into the environment originates as a byproduct of coal burning and other industrial processes. It
A physician who has a group of thirty-eight female patients aged 18 to 24 on a special diet wishes to estimate the effect of the diet on total serum cholesterol. For this group, their average serum
Suppose a sample of size n is to be drawn from a normal distribution where σ is known to be 14.3. How large does n have to be to guarantee that the length of the 95% confidence interval for μ will
What "confidence" would be associated with each of the following intervals? Assume that the random variable Y is normally distributed and that σ is known.(a) (y̅− 1.64・σ/√n, y̅ +
Five independent samples, each of size n, are to be drawn from a normal distribution where σ is known. For each sample, the interval (y̅ − 0.96・σ√n, y̅ + 1.06・σ√n) will be constructed.
Two chips are drawn without replacement from an urn containing five chips, numbered 1 through 5. The average of the two drawn is to be used as an estimator, ˆθ, for the true average of all the
A sample of size 1 is drawn from the uniform pdf defined over the interval [0, θ]. Find an unbiased estimator for θ2.
Suppose that W is an unbiased estimator for θ. Can W2 be an unbiased estimator for θ2?
We showed in Example 5.4.4 that ˆσ2 = 1/n is biased for σ2. Suppose μ is known and does not have to be estimated by Y̅. Show that ˆσ2 = is unbiased for σ2.
As an alternative to imposing unbiasedness, an estimator's distribution can be "centered" by requiring that its median be equal to the unknown parameter θ. If it is, ˆ θ is said to be median
Let Y1, Y2, . . . , Yn be a random sample of size n from the pdf fY(y; θ) = 1/θ e−y/θ, y > 0. Let ˆθ = n・ Ymin. Is ˆθ unbiased for θ? Is ˆθ = unbiased for θ?
An estimator ˆθn = h(W1, . . . ,Wn) is said to be asymptotically unbiased for θ if = θ. Suppose W is a random variable with E(W) = μ and with variance σ2. Show that W2 is an
Is the maximum likelihood estimator for σ2 in a normal pdf, where both μ and σ2 are unknown, asymptotically unbiased?
Let X1, X2, . . . , Xn denote the outcomes of a series of n independent trials, wherefor i =1, 2, . . . , n. Let X = X1 + X2 +・ ・ ・+ Xn.(a) Show that p̂1 = X1 and p̂2 = X/n are unbiased
Suppose that n = 5 observations are taken from the uniform pdf, fY (y; θ) = 1/θ, 0 ≤ y ≤ θ, where θ is unknown. Two unbiased estimators for θ are ˆθ1 = 6/5・Ymax and ˆθ2 = 6・YminWhich
Let Y1, Y2, . . . , Yn be a random sample of size n from the pdf fY(y; θ) = 1 θ e−y/θ, y > 0.(a) Show that ˆθ1 = Y1, ˆθ2 = Y, and ˆθ3 = n・ Ymin are all unbiased estimators for θ.(b)
Suppose a random sample of size n = 6 is drawn from the uniform pdf fY(y; θ) = 1/θ, 0 ≤ y ≤ θ, for the purpose of using ˆθ =Ymax to estimate θ.(a) Calculate the probability that ˆ θ falls
Given a random sample of size n from a Poisson distribution, ˆλ1 = X1 and ˆλ2 = X are two unbiased estimators for λ. Calculate the relative efficiency of ˆλ1 to ˆλ2.
If Y1, Y2, . . . , Yn are random observations from a uniform pdf over [0, θ], both ˆθ1 = (n + 1/n)・Ymax and ˆθ2 = (n + 1). Ymin are unbiased estimators for θ. Show that Var(ˆθ2)/Var(ˆθ1)
Suppose that W1 is a random variable with mean μ and variance σ21 and W2 is a random variable with mean μ and variance σ22. From Example 5.4.3, we know that cW1 + (1 − c)W2 is an unbiased
Five hundred adults are asked whether they favor a bipartisan campaign finance reform bill. If the true proportion of the electorate in favor of the legislation is 52%, what are the chances that
A sample of size n = 16 is drawn from a normal distribution where σ = 10 but μ is unknown. If μ = 20, what is the probability that the estimator ˆμ = Y will lie between 19.0 and 21.0?
Suppose X1, X2, . . . , Xn is a random sample of size n drawn from a Poisson pdf where λ is an unknown parameter. Show that ˆλ = X is unbiased for λ. For what type of parameter, in general, will
Let Ymin be the smallest order statistic in a random sample of size n drawn from the uniform pdf, fY(y; θ) = 1/θ, 0 ≤ y ≤ θ. Find an unbiased estimator for θ based on Ymin.
Let Y be the random variable described in Example 5.2.3, where fY (y, θ) = e−(y−θ), y ≥ θ, θ > 0. Show that Ymin – 1/n is an unbiased estimator of θ.
Suppose that 14, 10, 18, and 21 constitute a random sample of size 4 drawn from a uniform pdf defined over the interval [0, θ], where θ is unknown. Find an unbiased estimator for θ based on Yʹ3,
A random sample of size 2, Y1 and Y2, is drawn from the pdffY(y; θ) = 2yθ2, 0 < y < 1/θWhat must c equal if the statistic c(Y1 + 2Y2) is to be an unbiased estimator for 1/θ?
Let Y1, Y2, . . . , Yn be a random sample from fY(y; θ) = 1/θ e−y/θ, y > 0. Compare the Cramér-Rao lower bound for fY(y; θ) to the variance of the maximum likelihood estimator for θ, ˆθ
Let X1, X2, . . . , Xn be a random sample of size n from the Poisson distribution, pX(k; λ) = e−λλk/k!, k = 0, 1, . . .. Show that ˆλ = Xi is an efficient estimator for λ.
Suppose a random sample of size n is taken from a normal distribution with mean μ and variance σ2, where σ2 is known. Compare the Cramér-Rao lower bound for fY(y; μ) with the variance of ˆμ =
Let Y1, Y2, . . . , Yn be a random sample from the uniform pdf fY(y; θ) = 1/θ, 0 ≤ y ≤ θ. Compare the Cramér-Rao lower bound for fY(y; θ) with the variance of the unbiased estimator ˆθ = n
Let X have the pdf fX(k; θ) = (θ − 1)k−1/θk, k = 1, 2, 3, . . ., θ >1, which is geometric (p = 1/θ). For this pdf E(X) = θ and Var(X) = θ(θ −1). Is the statistic X efficient?
Let Y1, Y2, . . . , Yn be a random sample of size n from the pdffY(y; θ) = 1 (r −1)!θr yr−1e−y/θ, y > 0(a) Show that ˆθ = 1/r Y̅ is an unbiased estimator for θ.(b) Show that ˆθ =
Prove the equivalence of the two forms given for the Cramér-Rao lower bound in Theorem 5.5.1. [Differentiate the equation with respect to θ and deduce that & Then differentiate again with
Let X1, X2, . . . , Xn be a random sample of size n from the geometric distribution, pX(k; p) = (1 − p)k−1 p, k = 1, 2, . . .. Show that p̂ = is sufficient for p.
Write the pdf fY(y; λ) = λe−λy, y > 0, in exponential form and deduce a sufficient statistic for λ. Assume that the data consist of a random sample of size n.
Let Y1, Y2, . . . , Yn be a random sample from a Pareto pdf,fY(y; θ) = θ/(1 + y)θ+1, 0 ≤ y ≤ ∞; 0 < θ < ∞Write fY(y; θ) in exponential form and deduce a sufficient statistic for θ
Let X1, X2, and X3 be a set of three independent Bernoulli random variables with unknown parameter p = P(Xi = 1). It was shown on p. 324 that p̂ = X1 + X2 + X3 is sufficient for p. Show that
If ˆθ is sufficient for θ, show that any one-to-one function of ˆθ is also sufficient for θ.
Show that is sufficient for σ2 if Y1, Y2, . . . , Yn is a random sample from a normal pdf with μ = 0.
Let Y1, Y2, . . . , Yn be a random sample of size n from the pdf of Question 5.5.6,fY(y; θ) = 1/(r − 1)!θr yr−1e−y/θ, 0 ≤ yfor positive parameter θ and r a known positive integer. Find a
Let Y1, Y2, . . . , Yn be a random sample of size n from the pdf fY(y; θ) = θ yθ−1, 0 ≤ y ≤ 1. Use Theorem 5.6.1 to show that W = is a sufficient statistic for θ. Is the maximum likelihood
Suppose a random sample of size n is drawn from the pdffY(y; θ) = e−(y−θ), θ ≤ y(a) Show that ˆθ = Ymin is sufficient for the threshold parameter θ.(b) Show that Ymax is not sufficient
Suppose a random sample of size n is drawn from the pdffY(y; θ) = 1/θ , 0 ≤ y ≤ θFind a sufficient statistic for θ.
A probability model gW(w; θ) is said to be expressed in exponential form if it can be written asgW(w; θ) = eK(w)p(θ)+S(w)+q(θ)where the range of W is independent of θ. Show that K(Wi) is
How large a sample must be taken from a normal pdf where E(Y) = 18 in order to guarantee that has a 90% probability of lying somewhere in the interval [16, 20]? Assume that σ = 5.0.
Let Y1, Y2, . . . , Yn be a random sample of size n from a normal pdf having μ = 0. Show that S2n = is a consistent estimator for σ2 = Var(Y).
Suppose Y1, Y2, . . . , Yn is a random sample from the exponential pdf, fY(y; λ) = λe−λy, y > 0.(a) Show that ˆλn = Y1 is not consistent for λ.(b) Show that ˆλn = is not consistent
An estimator ˆθn is said to be squared-error consistent for θ if(a) Show that any squared-error consistent ˆθn is asymptotically unbiased (see Question 5.4.15).(b) Show that any squared-error
Suppose ˆθn = Ymax is to be used as an estimator for the parameter θ in the uniform pdf, fY(y; θ) = 1/θ, 0 ≤ y ≤ θ. Show that ˆθn is squared-error consistent
If 2n + 1 random observations are drawn from a continuous and symmetric pdf with mean μ and if fY(μ; μ) = 0, then the sample median, Yʹn+1, is unbiased for μ, and Var(Yʹn+1) = 1/(8[fY(μ;
Suppose that X is a geometric random variable, where pX(k|θ) = (1−θ)k−1θ, k = 1, 2, . . . . Assume that the prior distribution for θ is the beta pdf with parameters r and s. Find the
Find the squared-error loss [L(ˆθ, θ) = (ˆθ − θ)2] Bayes estimate for θ in Example 5.8.2 and express it as a weighted average of the maximum likelihood estimate for θ and the mean of the
Suppose the binomial pdf described in Example 5.8.2 refers to the number of votes a candidate might receive in a poll conducted before the general election. Moreover, suppose a beta prior
In Questions 5.8.2-5.8.4, is the Bayes estimate unbiased? Is it asymptotically unbiased?
Suppose that Y is a gamma random variable with parameters r and θ and the prior is also gamma with parameters s and μ. Show that the posterior pdf is gamma with parameters r +s and y + μ.
Let Y1, Y2, . . . , Yn be a random sample from a gamma pdf with parameters r and θ, where the prior distribution assigned to θ is the gamma pdf with parameters s and μ. Let W = Y1 + Y2 +・ ・
Consider, again, the scenario described in Example 5.8.2-a binomial random variable X has parameters n and θ, where the latter has a beta prior with integer parameters r and s. Integrate the joint
State the decision rule that would be used to test the following hypotheses. Evaluate the appropriate test statistic and state your conclusion.(a) H0: μ = 120 versus H1: μ < 120; y̅ = 114.2, n
As a class research project, Rosaura wants to see whether the stress of final exams elevates the blood pressures of freshmen women. When they are not under any untoward duress, healthy
As input for a new inflation model, economists predicted that the average cost of a hypothetical "food basket" in east Tennessee in July would be $145.75. The standard deviation (σ) of basket prices
An herbalist is experimenting with juices extracted from berries and roots that may have the ability to affect the Stanford-Binet IQ scores of students afflicted with mild cases of attention deficit
(a) Suppose H0: μ = μo is rejected in favor of H1: μ = μo at the α = 0.05 level of significance. Would H0 necessarily be rejected at the α = 0.01 level of significance?(b) Suppose H0: μ = μo
Company records show that drivers get an average of 32,500 miles on a set of Road Hugger All-Weather radial tires. Hoping to improve that figure, the company has added a new polymer to the rubber
A random sample of size 16 is drawn from a normal distribution having σ =6.0 for the purpose of testing H0: μ = 30 versus H1: μ = 30. The experimenter chooses to define the critical region C to be
Recall the breath analyzers described in Example 4.3.5. The following are thirty blood alcohol determinations made by Analyzer GTE-10, a three-year-old unit that may be in need of recalibration. All
Calculate the P-values for the hypothesis tests indicated in Question 6.2.1. Do they agree with your decisions on whether or not to reject H0?(a) H0: μ = 120 versus H1: μ < 120; y̅ = 114.2, n =
Suppose H0: μ = 120 is tested against H1: μ ≠ 120. If σ = 10 and n = 16, what P-value is associated with the sample mean y =122.3? Under what circumstances would H0 be rejected?
Commercial fishermen working certain parts of the Atlantic Ocean sometimes find their efforts hindered by the presence of whales. Ideally, they would like to scare away the whales without frightening
Efforts to find a genetic explanation for why certain people are right-handed and others left-handed have been largely unsuccessful. Reliable data are difficult to find because of environmental
Defeated in his most recent attempt to win a congressional seat because of a sizeable gender gap, a politician has spent the last two years speaking out in favor of women's rights issues. A newly
Suppose H0: p = 0.45 is to be tested against H1: p > 0.45 at the α = 0.14 level of significance, where p = P(ith trial ends in success). If the sample size is 200, what is the smallest number of
Recall the median test described in Example 5.3.2. Reformulate that analysis as a hypothesis test rather than a confidence interval. What P-value is associated with the outcomes listed in Table 5.3.3?
Among the early attempts to revisit the death postponement theory introduced in Case Study 6.3.2 was an examination of the birth dates and death dates of 348 U.S. celebrities (134). It was found that
What α levels are possible with a decision rule of the form "Reject H0 if k ≥ k∗" when H0: p = 0.5 is to be tested against H1: p > 0.5 using a random sample of size n =7?
The following is a Minitab printout of the binomial pdf pX(k) = (9/k) (0.6)k(0.4)9−k, k = 0, 1, . . . , 9. Suppose H0: p = 0.6 is to be tested against H1: p > 0.6 and we wish the level of
Suppose H0: p = 0.75 is to be tested against H1: p < 0.75 using a random sample of size n =7 and the decision rule "Reject H0 if k ≤ 3."(a) What is the test's level of significance?(b) Graph the
Recall the "Math for the Twenty-First Century" hypothesis test done in Example 6.2.1. Calculate the power of that test when the true mean is 500.
Suppose a sample of size 1 is taken from the pdf fY(y) = (1/λ)e−y/λ, y > 0, for the purpose of testingH0: λ = 1versusH1: λ > 1The null hypothesis will be rejected if y ≥ 3.20.(a)
Polygraphs used in criminal investigations typically measure five bodily functions: (1) thoracic respiration, (2) abdominal respiration, (3) blood pressure and pulse rate, (4) muscular movement and
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