Suppose that X1, . . . , Xn are normal with mean 1; Y1, .

Let X1, . . . , Xn be a sample from the distribution whose density function is f (x) = _e(x) x 0 otherwise Determine the maximum likelihood estimator of .

Determine the maximum likelihood estimator of when X1, . . . , Xn is a sample with density function f (x) = 12 e|x|, < x <

Let X1, . . . , Xn be a sample from a normal , 2 population. Determine the maximum likelihood estimator of 2 when is known. What is the expected value of this estimator?

Determine the maximum likelihood estimates of a and when X1, . . . , Xn is a sample from the Pareto density function f (x) = _ ax(+1), if x a 0, if x < a

Suppose that X1, . . . , Xn are normal with mean 1; Y1, . . . , Yn are normal with mean 2; and W1, . . . ,Wn are normal with mean 1 +2. Assuming that all 3n random variables are independent with a common variance, find the maximum likelihood estimators of 1 and 2.

River floods are often measured by their discharges (in units of feet cubed per second). The value v is said to be the value of a 100-year flood if P{D v} = .01 where D is the discharge of the largest flood in a randomly chosen year. The following table gives the flood discharges of the largest floods of the Blackstone River in Woonsocket, Rhode Island, in each of the years from 1929 to 1965. Assuming that these discharges follow a lognormal distribution, estimate the value of a 100-year flood.

Recall that X is said to have a lognormal distribution with parameters and 2 if log(X) is normal with mean and variance 2. Suppose X is such a lognormal random variable. (a) Find E[X]. (b) Find Var(X). Hint: Make use of the formula for the moment generating function of a normal random variable. (c) The following are, in minutes, travel times to work over a sequence of 10 days. 42, 28, 53, 57, 67, 39, 35, 50, 44, 39 Assuming an underlying lognormal distribution, use the data to estimate themean travel time.

An electric scale gives a reading equal to the true weight plus a random error that is normally distributed with mean 0 and standard deviation = .1 mg. Suppose that the results of five successive weighings of the same object are as follows: 3.142, 3.163, 3.155, 3.150, 3.141. (a) Determine a 95 percent confidence interval estimate of the true weight. (b) Determine a 99 percent confidence interval estimate of the true weight.

The PCB concentration of a fish caught in Lake Michigan was measured by a technique that is known to result in an error of measurement that is normally distributed with a standard deviation of .08 ppm (parts per million). Suppose the results of 10 independent measurements of this fish are 11.2, 12.4, 10.8, 11.6, 12.5, 10.1, 11.0, 12.2, 12.4, 10.6 (a) Give a 95 percent confidence interval for the PCB level of this fish. (b) Give a 95 percent lower confidence interval. (c) Give a 95 percent upper confidence interval.

The standard deviation of test scores on a certain achievement test is 11.3. If a random sample of 81 students had a sample mean score of 74.6, find a 90 percent confidence interval estimate for the average score of all students.

Let X1, . . . , Xn, Xn+1 be a sample from a normal population having an unknown mean and variance 1. Let .Xn = _ni =1 Xi/n be the average of the first n of them. (a) What is the distribution of Xn+1 .Xn? (b) If . Xn = 4, give an interval that, with 90 percent confidence, will contain the value of Xn+1.

If X1, . . . , Xn is a sample from a normal population whose mean is unknown but whose variance 2 is known, show that (, X + z/ n) is a 100(1 ) percent lower confidence interval for .

A sample of 20 cigarettes is tested to determine nicotine content and the average value observed was 1.2 mg. Compute a 99 percent two-sided confidence interval for the mean nicotine content of a cigarette if it is known that the standard deviation of a cigarettes nicotine content is = .2 mg.

In Problem 13, suppose that the population variance is not known in advance of the experiment. If the sample variance is .04, compute a 99 percent two-sided confidence interval for the mean nicotine content.

In Problem 14, compute a value c for which we can assert with 99 percent confidence that c is larger than the mean nicotine content of a cigarette.

Suppose that when sampling from a normal population having an unknown mean and unknown variance 2, we wish to determine a sample size n so as to guarantee that the resulting 100(1 ) percent confidence interval for will be of size no greater than A, for given values and A. Explain how we can approximately do this by a double sampling scheme that first takes a subsample of size 30 and then chooses the total sample size by using the results of the first subsample.

The following data resulted from 24 independent measurements of the melting point of lead. 330C 322C 345C 328.6C 331C 342C 342.4C 340.4C 329.7C 334C 326.5C 325.8C 337.5C 327.3C 322.6C 341C 340C 333C 343.3C 331C 341C 329.5C 332.3C 340C Assuming that the measurements can be regarded as constituting a normal sample whose mean is the true melting point of lead, determine a 95 percent two-sided confidence interval for this value. Also determine a 99 percent two-sided confidence interval.

The following are scores on IQ tests of a random sample of 18 students at a large eastern university. 130, 122, 119, 142, 136, 127, 120, 152, 141, 132, 127, 118, 150, 141, 133, 137, 129, 142 (a) Construct a 95 percent confidence interval estimate of the average IQ score of all students at the university. (b) Construct a 95 percent lower confidence interval estimate. (c) Construct a 95 percent upper confidence interval estimate.

Suppose that a random sample of nine recently sold houses in a certain city resulted in a sample mean price of $222,000, with a sample standard deviation of $22,000. Give a 95 percent upper confidence interval for the mean price of all recently sold houses in this city.

A company self-insures its large fleet of cars against collisions. To determine its mean repair cost per collision, it has randomly chosen a sample of 16 accidents. If the average repair cost in these accidents is $2,200 with a sample standard deviation of $800, find a 90 percent confidence interval estimate of the mean cost per collision.

A standardized test is given annually to all sixth-grade students in the state of Washington. To determine the average score of students in her district, a school supervisor selects a random sample of 100 students. If the sample mean of these students scores is 320 and the sample standard deviation is 16, give a 95 percent confidence interval estimate of the average score of students in that supervisors district.

Each of 20 science students independently measured the melting point of lead. The sample mean and sample standard deviation of these measurements were (in degrees centigrade) 330.2 and 15.4, respectively. Construct (a) a 95 percent and (b) a 99 percent confidence interval estimate of the true melting point of lead.

A random sample of 300 CitiBank VISA cardholder accounts indicated a sample mean debt of $1,220 with a sample standard deviation of $840. Construct a 95 percent confidence interval estimate of the average debt of all cardholders.

In Problem 23, find the smallest value v that with 90 percent confidence, exceeds the average debt per cardholder.

Verify the formula given in Table 7.1 for the 100(1) percent lower confidence interval for when is unknown.

The following are the daily number of steps taken by a certain individual in 20 weekdays. 2,100 1,984 2,072 1,898 1,950 1,992 2,096 2,103 2,043 2,218 2,244 2,206 2,210 2,152 1,962 2,007 2,018 2,106 1,938 1,956 Assuming that the daily number of steps is normally distributed, construct (a) a 95 percent and (b) a 99 percent two-sided confidence interval for the mean number of steps. (c) Determine the largest value v that, with 95 percent confidence, will be less than the mean range.

Studies were conducted in Los Angeles to determine the carbon monoxide concentration near freeways. The basic technique used was to capture air samples in special bags and to then determine the carbon monoxide concentration by using a spectrophotometer. The measurements in ppm (parts per million) over a sampled period during the year were 102.2, 98.4, 104.1, 101, 102.2, 100.4, 98.6, 88.2, 78.8, 83, 84.7, 94.8, 105.1, 106.2, 111.2, 108.3, 105.2, 103.2, 99, 98.8. Compute a 95 percent two-sided confidence interval for the mean carbon monoxide concentration.

A set of 10 determinations, by a method devised by the chemist Karl Fischer, of the percentage of water in a methanol solution yielded the following data. .50, .55, .53, .56, .54, .57, .52, .60, .55, .58 Assuming normality, use these data to give a 95 percent confidence interval for the actual percentage.

Suppose that U1,U2, . . . is a sequence of independent uniform (0,1) random variables, and define N by N = min{n : U1 + +Un > 1} That is, N is the number of uniform (0, 1) random variables that need to be summed to exceed 1. Use random numbers to determine the value of 36 random variables having the same distribution as N, and then use these data to obtain a 95 percent confidence interval estimate of E[N]. Based on this interval, guess the exact value of E[N].

An important issue for a retailer is to decide when to reorder stock from a supplier. A common policy used to make the decision is of a type called s, S: The retailer orders at the end of a period if the on-hand stock is less than s, and orders enough to bring the stock up to S. The appropriate values of s and S depend on different cost parameters, such as inventory holding costs and the profit per item sold, as well as the distribution of the demand during a period. Consequently, it is important for the retailer to collect data relating to the parameters of the demand distribution. Suppose that the following data give the numbers of a certain type of item sold in each of 30 weeks. 14, 8, 12, 9, 5, 22, 15, 12, 16, 7, 10, 9, 15, 15, 12, 9, 11, 16, 8, 7, 15, 13, 9, 5, 18, 14, 10, 13, 7, 11 Assuming that the numbers sold each week are independent random variables from a common distribution, use the data to obtain a 95 percent confidence interval for the mean number sold in a week.

A random sample of 16 professors at a large private university yielded a sample mean annual salary of $90,450 with a sample standard deviation of $9,400. Determine a 95 percent confidence interval of the average salary of all professors at that university.

Let X1, . . . , Xn+1 be a sample from a population with mean and variance 2. As noted in the text, the natural predictor of Xn+1 based on the data values X1, . . . , Xn is .Xn = _ni =1 Xi/n. Determine the mean square error of this predictor. That is, find E[(Xn+1 .Xn)2].

National Safety Council data show that the number of accidental deaths due to drowning in the United States in the years from 1990 to 1993 were (in units of one thousand) 5.2, 4.6, 4.3, 4.8. Use these data to give an interval that will, with 95 percent confidence, contain the number of such deaths in 1994.

The daily dissolved oxygen concentration for a water stream has been recorded over 30 days. If the sample average of the 30 values is 2.5 mg/liter and the sample standard deviation is 2.12 mg/liter, determine a value which, with 90 percent confidence, exceeds the mean daily concentration.

Verify the formulas given in Table 7.1 for the 100(1 ) percent lower and upper confidence intervals for 2.

The capacities (in ampere-hours) of 10 batteries were recorded as follows: 140, 136, 150, 144, 148, 152, 138, 141, 143, 151 (a) Estimate the population variance 2. (b) Compute a 99 percent two-sided confidence interval for 2. (c) Compute a value v that enables us to state, with 90 percent confidence, that 2 is less than v.

Find a 95 percent two-sided confidence interval for the variance of the diameter of a rivet based on the data given here. 6.68 6.66 6.62 6.72 6.76 6.67 6.70 6.72 6.78 6.66 6.76 6.72 6.76 6.70 6.76 6.76 6.74 6.74 6.81 6.66 6.64 6.79 6.72 6.82 6.81 6.77 6.60 6.72 6.74 6.70 6.64 6.78 6.70 6.70 6.75 6.79 Assume a normal population.

The following are independent samples from two normal populations, both of which have the same standard deviation . 16, 17, 19, 20, 18 and 3, 4, 8 Use them to estimate .

The amount of beryllium in a substance is often determined by the use of a photometric filtration method. If the weight of the beryllium is , then the value given by the photometric filtration method is normally distributed with mean and standard deviation . A total of eight independent measurements of 3.180 mg of beryllium gave the following results. 3.166, 3.192, 3.175, 3.180, 3.182, 3.171, 3.184, 3.177 Use the preceding data to (a) estimate ; (b) find a 90 percent confidence interval estimate of .

If X1, . . . , Xn is a sample from a normal population, explain how to obtain a 100(1 ) percent confidence interval for the population variance 2 when the population mean is known. Explain in what sense knowledge of improves the interval estimator compared with when it is unknown. Repeat Problem 38 if it is known that the mean burning time is 53.6 seconds.

A civil engineer wishes to measure the compressive strength of two different types of concrete. A random sample of 10 specimens of the first type yielded the following data (in psi) Type 1: 3,250, 3,268, 4,302, 3,184, 3,266 3,297, 3,332, 3,502, 3,064, 3,116 whereas a sample of 10 specimens of the second yielded the data Type 2: 3,094, 3,106, 3,004, 3,066, 2,984, 3,124, 3,316, 3,212, 3,380, 3,018 If we assume that the samples are normal with a common variance, determine (a) a 95 percent two-sided confidence interval for 1 2, the difference in means; (b) a 95 percent one-sided upper confidence interval for 1 2; (c) a 95 percent one-sided lower confidence interval for 1 2.

Independent random samples are taken from the output of two machines on a production line. The weight of each item is of interest. From the first machine, a sample of size 36 is taken, with sample mean weight of 120 grams and a sample variance of 4. From the second machine, a sample of size 64 is taken, with a sample mean weight of 130 grams and a sample variance of 5. It is assumed that the weights of items from the first machine are normally distributed with mean 1 and variance 2 and that the weights of items from the second machine are normally distributed with mean 2 and variance 2 (that is, the variances are assumed to be equal). Find a 99 percent confidence interval for 1 2, the difference in population means.

Do Problem 42 when it is known in advance that the population variances are 4 and 5.

The following are the daily numbers of company website visits resulting from advertisements on two different types of media. Type I Type II 481 572 526 537 506 561 511 582 527 501 556 605 661 487 542 558 501 524 491 578 Find a 99 percent confidence interval for the mean difference in daily visits assuming normality with unknown but equal variances.

If X1, . . . , Xn is a sample from a normal population having known mean 1 and unknown variance 2 1 , and Y1, . . . , Ym is an independent sample from a normal population having known mean 2 and unknown variance 2 2 , determine a 100(1 ) percent confidence interval for 2 1 /2 2 .

Two analysts took repeated readings on the hardness of city water. Assuming that the readings of analyst i constitute a sample from a normal population having variance 2 i , i = 1, 2, compute a 95 percent two-sided confidence interval for 2 1 /2 2 when the data are as follows: Coded Measures of Hardness Analyst 1 Analyst 2 .46 .82 .62 .61 .37 .89 .40 .51 .44 .33 .58 .48 .48 .23 .53 .25 .67 .88

A problem of interest in baseball is whether a sacrifice bunt is a good strategy when there is a man on first base and no outs. Assuming that the bunter will be out but will be successful in advancing the man on base, we could compare the probability of scoring a run with a player on first base and no outs with the probability of scoring a run with a player on second base and one out. The following data resulted from a study of randomly chosen major league baseball games played in 1959 and 1960. (a) Give a 95 percent confidence interval estimate for the probability of scoring at least one run when there is a man on first and no outs. (b) Give a 95 percent confidence interval estimate for the probability of scoring at least one run when there is a man on second and one out. Number of Cases Number in Which 0 Runs Total Number Base Occupied of Outs Are Scored of Cases First 0 1,044 1,728 Second 1 401 657

A random sample of 1,200 engineers included 48 Hispanic Americans, 80 African Americans, and 204 females. Determine 90 percent confidence intervals for the proportion of all engineers who are (a) female; (b) Hispanic Americans or African Americans.

To estimate p, the proportion of all newborn babies that are male, the gender of 10,000 newborn babies was noted. If 5,106 of them were male, determine (a) a 90 percent and (b) a 99 percent confidence interval estimate of p.

An airline is interested in determining the proportion of its customers who are flying for reasons of business. If they want to be 90 percent certain that their estimate will be correct to within 2 percent, how large a random sample should they select?

A recent newspaper poll indicated that Candidate A is favored over Candidate B by a 53 to 47 percentage, with a margin of error of 4 percent. The newspaper then stated that since the 6-point gap is larger than the margin of error, its readers can be certain that Candidate A is the current choice. Is this reasoning correct?

A market research firm is interested in determining the proportion of households that are watching a particular sporting event. To accomplish this task, they plan on using a telephone poll of randomly chosen households. How large a sample is needed if they want to be 90 percent certain that their estimate is correct to within .02?

In a recent study, 79 of 140 meteorites were observed to enter the atmosphere with a velocity of less than 25 miles per second. If we take p = 79/140 as an estimate of the probability that an arbitrary meteorite that enters the atmosphere will have a speed less than 25 miles per second, what can we say, with 99 percent confidence, about the maximum error of our estimate? Problems 295

A random sample of 100 items from a production line revealed 17 of them to be defective. Compute a 95 percent two-sided confidence interval for the probability that an item produced is defective. Determine also a 99 percent upper confidence interval for this value. What assumptions are you making?

Of 100 randomly detected cases of individuals having lung cancer, 67 died within 5 years of detection. (a) Estimate the probability that a person contracting lung cancer will die within 5 years. (b) How large an additional sample would be required to be 95 percent confident that the error in estimating the probability in part (a) is less than .02?

Derive 100(1 ) percent lower and upper confidence intervals for p, when the data consist of the values of n independent Bernoulli random variables with parameter p.

Suppose the lifetimes of batteries are exponentially distributed with mean . If the average of a sample of 10 batteries is 36 hours, determine a 95 percent two-sided confidence interval for .

Determine 100(1 ) percent one-sided upper and lower confidence intervals for in Problem 57.

Let X1, X2, . . . , Xn denote a sample from a population whose mean value is unknown. Use the results of Example 7.7b to argue that among all unbiased estimators of of the form _ni =1 iXi ,_ni =1 i = 1, the one with minimal mean square error has i 1/n, i = 1, . . . , n.

Consider two independent samples from normal populations having the same variance 2, of respective sizes n and m. That is, X1, . . . , Xn and Y1, . . . , Ym are independent samples from normal populations each having variance 2. Let S2 x and S2 y denote the respective sample variances. Thus both S2 x and S2 y are unbiased estimators of 2. Show by using the results of Example 7.7b along with the fact that Var(2 k ) = 2k where 2 k is chi-square with k degrees of freedom, that the minimum mean square estimator of 2 of the form S2 x + (1 )S2 y is S2 p = (n 1)S2 x + (m 1)S2 y n + m 2 This is called the pooled estimator of 2.

Consider two estimators d1 and d2 of a parameter . If E[d1] = , Var(d1) = 6 and E[d2] = + 2, Var(d2) = 2, which estimator should be preferred?

Suppose that the number of accidents occurring daily in a certain plant has a Poisson distribution with an unknown mean . Based on previous experience in similar industrial plants, suppose that a statisticians initial feelings about the possible value of can be expressed by an exponential distribution with parameter 1. That is, the prior density is p() = e, 0< < Determine the Bayes estimate of if there are a total of 83 accidents over the next 10 days. What is the maximum likelihood estimate?

The functional lifetimes in hours of computer chips produced by a certain semiconductor firm are exponentially distributed with mean 1/. Suppose that the prior distribution on is the gamma distribution with density function g(x) = exx2 2 , 0< x < If the average life of the first 20 chips tested is 4.6 hours, compute the Bayes estimate of .

Each item produced will, independently, be defective with probability p. If the prior distribution on p is uniform on (0, 1), compute the posterior probability that p is less than .2 given (a) a total of 2 defectives out of a sample of size 10; (b) a total of 1 defective out of a sample of size 10; (c) a total of 10 defectives out of a sample of size 10.

The breaking strength of a certain type of cloth is to be measured for 10 specimens. The underlying distribution is normal with unknown mean but with a standard deviation equal to 3 psi. Suppose also that based on previous experience we feel that the unknown mean has a prior distribution that is normally distributed with mean 200 and standard deviation 2. If the average breaking strength of a sample of 20 specimens is 182 psi, determine a region that contains with probability .95.