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Questions and Answers of Statistics

An engineer who is studying the tensile strength of a steel alloy intended for use in golf club shafts knows that tensile strength is approximately normally distributed with σ = 60 psi. A random
Suppose that in Exercise 9-25 we wanted to reject the null hypothesis with probability at least 0.8 if mean strength μ = 3500. What sample size should be used?
Supercavitation is a propulsion technology for undersea vehicles that can greatly increase their speed. It occurs above approximately 50 meters per second, when pressure drops sufficiently to allow
A bearing used in an automotive application is suppose to have a nominal inside diameter of 1.5 inches. A random sample of 25 bearings is selected and the average inside diameter of these bearings is
Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient’s
An article in the ASCE Journal of Energy Engineering (1999, Vol. 125, pp. 59–75) describes a study of the thermal inertia properties of autoclaved aerated concrete used as a building material. Five
A 1992 article in the Journal of the American Medical Association (“A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold
Cloud seeding has been studied for many decades as a weather modification procedure (for an interesting study of this subject, see the article in Techno metrics by Simpson, Alsen, and Eden, “A
The sodium content of thirty 300-gram boxes of organic corn flakes was determined. The data (in milligrams) are as follows: 131.15, 130.69, 130.91, 129.54, 129.64, 128.77, 130.72, 128.33, 128.24,
Reconsider the tire testing experiment described in Exercise 8-22. (a) The engineer would like to demonstrate that the mean life of this new tire is in excess of 60,000 kilometers. Formulate and
Reconsider the Izod impact test on PVC pipe described in Exercise 8-23. Suppose that you want to use the data from this experiment to support a claim that the mean impact strength exceeds the ASTM
Reconsider the television tube brightness experiment in Exercise 8-24. Suppose that the design engineer believes that this tube will require 300 micro amps of current to produce the desired
Consider the baseball coefficient of restitution data first presented in Exercise 8-79.(a) Does the data support the claim that the mean coefficient of restitution of baseballs exceeds 0.635? Use a =
Consider the dissolved oxygen concentration at TVA dams first presented in Exercise 8-81. (a) Test the hypotheses H0: μ = 4 versus H1 μ  4 Use a = 0.01. (b) What is the P-value of
Consider the cigar tar content data first presented in Exercise 8-82.(a) Can you support a claim that mean tar content exceeds 1.5? Use a = 0.05(b) What is the P-value of the test statistic computed
Exercise 6-22 gave data on the heights of female engineering students at ASU.(a) Can you support a claim that mean height of female engineering students at ASU is 65 inches? Use a = 0.05(b) What is
Exercise 6-24 presented data on the concentration of suspended solids in lake water. (a) Test the hypotheses H0: μ = 55 versus, use μ = 0.05. (b) What is the P-value of the test
Exercise 6-25 describes testing golf balls for an overall distance standard.(a) Can you support a claim that means distance achieved by this particular golf ball exceeds 280 yards? Use a = 0.05.(b)
Consider the rivet holes from Exercise 8-35. If the standard deviation of hole diameter exceeds 0.01 millimeters, there is an unacceptably high probability that the rivet will not fit. Recall that n
Recall the sugar content of the syrup in canned peaches from Exercise 8-36. Suppose that the variance is thought to be 2 = 18 (milligrams)2. A random sample of n = 10 cans yields a sample standard
Consider the tire life data in Exercise 8-22.(a) Can you conclude, using a = 0.05, that the standard deviation of tire life exceeds 200 kilometers? State any necessary assumptions about the
Consider the Izod impact test data in Exercise 8-23. (a) Test the hypothesis that σ = 0.10 against an alternative specifying that σ = 0.10, using a = 0.01, and draw a conclusion. State
Reconsider the percentage of titanium in an alloy used in aerospace castings from Exercise 8-39. Recall that s = 0.37 and n = 51. (a) Test the hypothesis H0: σ = 0.25 versus H1: σ = 0.25
Consider the hole diameter data in Exercise 8-35. Suppose that the actual standard deviation of hole diameter exceeds the hypothesized value by 50%. What is the probability that this difference will
Consider the sugar content in Exercise 9-44. Suppose that the true variance is 2 = 40. How large a sample would be required to detect this difference with probability at least 0.90?
In a random sample of 85 automobile engine crankshaft bearings, 10 have a surface finish roughness that exceeds the specifications. Does this data present strong evidence that the proportion of
Continuation of Exercise 9-50. If it is really the situation that p = 0.15, how likely is it that the test procedure in Exercise 9-50 will not reject the null hypothesis? If p = 0.15, how large would
Reconsider the integrated circuits described in Exercise 8-48.(a) Use the data to test H0: p = 0.05 versus H1: p = 0.05. Use a = 0.05.(b) Find the P-value for the test.
Consider the defective circuit data in Exercise 8-48.(a) Do the data support the claim that the fraction of defective units produced is less than 0.05, using a = 0.05?(b) Find the P-value for the
An article in Fortune (September 21, 1992) claimed that nearly one-half of all engineers continue academic studies beyond the B.S. degree, ultimately receiving either an M.S. or a Ph.D. degree. Data
A manufacturer of interocular lenses is qualifying a new grinding machine and will qualify the machine if the percentage of polished lenses that contain surface defects does not exceed 2%. A random
A researcher claims that at least 10% of all football helmets have manufacturing flaws that could potentially cause injury to the wearer. A sample of 200 helmets revealed that 16 helmets contained
A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels year-round to reduce air pollution. If more than 315 voters respond positively, we will conclude
The advertised claim for batteries for cell phones is set at 48 operating hours, with proper charging procedures. A study of 5000 batteries is carried out and 15 stop operating prior to 48 hours. Do
Consider the following frequency table of observations on the random variable X.Values 0 1 2 3 4Observed Frequency 24 30 31 11 4(a) Based on these 100 observations, is a Poisson distribution with a
Let X denote the number of flaws observed on a large coil of galvanized steel. Seventy-five coils are inspected and the following data were observed for the values of X:Values 1 2 3 4 5 6 7
The number of calls arriving at a switchboard from noon to 1 PM during the business days Monday through Friday is monitored for six weeks (i.e., 30 days). Let X be defined as the number of calls
Consider the following frequency table of observations on the random variable X:Values 0 1 2 3 4 Frequency 4 21 10 13 2(a) Based on these 50 observations, is a binomial distribution with
Define X as the number of under filled bottles from a filling operation in a carton of 24 bottles. Sixty cartons are inspected and the following observations on X are recorded: Values 0 1 2
The number of cars passing eastbound through the intersection of Mill and University Avenues has been tabulated by a group of civil engineering students. They have obtained the data in the adjacent
A company operates four machines three shifts each day. From production records, the following data on the number of breakdowns are collected:Test the hypothesis (using a = 0.05) that breakdowns are
Patients in a hospital are classified as surgical or medical. A record is kept of the number of times patients require nursing service during the night and whether or not these patients are on
Grades in a statistics course and an operations research course taken simultaneously were as follows for a group of students.Are the grades in statistics and operations research related? Use a = 0.01
An experiment with artillery shells yields the following data on the characteristics of lateral deflections and ranges. Would you conclude that deflection and range are independent? Use a = 0.05.
A study is being made of the failures of an electronic component. There are four types of failures possible and two mounting positions for the device. The following data have been taken:Would you
A random sample of students is asked their opinions on a proposed core curriculum change. The results are as follows.Test the hypothesis that opinion on the change is independent of class standing.
A manufacturer of semiconductor devices takes a random sample of size n of chips and tests them, classifying each chip as defective or non-defective. Let Xi = 0 if the chip is non-defective and Xi =
(a) n = 50, ˆp = 0.095(b) n = 100, ˆp= 0.095(c) n = 500, ˆp = 0.095(d) n = 1000, ˆp = 0.095(e) Comment on the effect of sample size on the observed P-value of the test.
An inspector of flow metering devices used to administer fluid intravenously will perform a hypothesis test to determine whether the mean flow rate is different from the flow rate setting of 200
Suppose that in Exercise 9-73, the experimenter had believed that σ = 14. For each of the following sample sizes, and a fixed σ = 0.05, find the probability of a type II error if the true
The marketers of shampoo products know that customers like their product to have a lot of foam. A manufacturer of shampoo claims that the foam height of his product exceeds 200 millimeters. It is
Suppose we wish to test the hypothesis H0: μ = 85 versus the alternative H1: μ > 85 where σ = 16. Suppose that the true mean is μ = 86 and that in the practical context of the
The cooling system in a nuclear submarine consists of an assembly of welded pipes through which a coolant is circulated. Specifications require that weld strength must meet or exceed 150 psi. (a)
Suppose we are testing H0: p = 0.5 versus H0: p ( 0.5. Suppose that p is the true value of the population proportion.(a) Using a = 0.05, find the power of the test for n = 100, 150, and 300 assuming
Consider the television picture tube brightness experiment described in Exercise 8-24.(a) For the sample size n = 10, do the data support the claim that the standard deviation of current is less than
Consider the fatty acid measurements for the diet margarine described in Exercise 8-25. (a) For the sample size n = 6, using a two-sided alternative hypothesis and a = 0.01, test H0: σ2 =
A manufacturer of precision measuring instruments claims that the standard deviation in the use of the instruments is at most 0.00002 millimeter. An analyst, who is unaware of the claim, uses the
A biotechnology company produces a therapeutic drug whose concentration has a standard deviation of 4 grams per liter. A new method of producing this drug has been proposed, although some additional
Consider the 40 observations collected on the number of nonconforming coil springs in production batches of size 50 given in Exercise 6-79.(a) Based on the description of the random variable and
Consider the 20 observations collected on the number of errors in a string of 1000 bits of a communication channel given in Exercise 6-80.(a) Based on the description of the random variable and these
Consider the spot weld shear strength data in Exercise 6-23. Does the normal distribution seem to be a reasonable model for these data? Perform an appropriate goodness-of-fit test to answer this
Consider the water quality data in Exercise 6-24. (a) Do these data support the claim that mean concentration of suspended solids does not exceed 50 parts per million?Use a = 0.05.(b) What is the
Consider the golf ball overall distance data in Exercise 6-25.(a) Do these data support the claim that the mean overall distance for this brand of ball does not exceed 270 yards? Use a = 0.05.(b)
Consider the baseball coefficient of restitution data in Exercise 8-79. If the mean coefficient of restitution exceeds 0.635, the population of balls from which the sample has been taken will be too
Consider the dissolved oxygen data in Exercise 8-81. Water quality engineers are interested in knowing whether these data support a claim that mean dissolved oxygen concentration is 2.5 milligrams
The mean pull-off force of an adhesive used in manufacturing a connector for an automotive engine application should be at least 75 pounds. This adhesive will be used unless there is strong evidence
For Example 11.11, verify that the following matrix is the inverse of I − Q and hence is the fundamental matrix N.Find Nc and NR. Interpret the results.
In Example 11.8, make states 0 and 4 into absorbing states. Find the fundamental matrix N, and also Nc and NR, for the resulting absorbing chain. Interpret the results.
Three tanks fight a three-way duel. Tank A has probability 1/2 of destroying the tank at which it fires, tank B has probability 1/3 of destroying the tank at which it fires, and tank C has
Smith is in jail and has 3 dollars; he can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him. If Smith bets A dollars, he wins A dollars with probability .4 and
Consider the game of tennis when deuce is reached. If a player wins the next point, he has advantage. On the following point, he either wins the game or the game returns to deuce. Assume that for any
Show that in both Example 11.11 and the example just given, the probability of absorption in a state having genes of a particular type is equal to the proportion of genes of that type in the starting
(E. Brown6) Mary and John are playing the following game: They have a three-card deck marked with the numbers 1, 2, and 3 and a spinner with the numbers 1, 2, and 3 on it. The game begins by dealing
(Roberts7) A city is divided into 3 areas 1, 2, and 3. It is estimated that amounts u1, u2, and u3 of pollution are emitted each day from these three areas. A fraction qij of the pollution from
We can use the gambling interpretation given in Exercise 28 to find the expected number of tosses required to reach pattern B when we start with pattern A. To be a meaningful problem, we assume that
In Example 11.11, define f(i) to be the proportion of G genes in state i. Show that f is a harmonic function (see Exercise 27). Why does this show that the probability of being absorbed in state
It is raining in the Land of Oz. Determine a tree and a tree measure for the next three days’ weather. Find w(1),w(2), and w(3) and compare with the results obtained from P, P2, and P3.
Consider the following process. We have two coins, one of which is fair, and the other of which has heads on both sides. We give these two coins to our friend, who chooses one of them at random (each
Consider a random walker who moves on the integers 0, 1, . . . , N, moving one step to the right with probability p and one step to the left with probability q = 1 − p. If the walker ever
Consider the Markov chain with general 2 Ã— 2 transition matrix(a) Under what conditions is P absorbing?(b) Under what conditions is P ergodic but not regular?(c) Under what conditions is
Prove that, if a 3-by-3 transition matrix has the property that its column sums are 1, then (1/3, 1/3, 1/3) is a fixed probability vector. State a similar result for n-by-n transition matrices.
Is the Markov chain in Example 11.11 ergodic?
For Example 11.4 when P is ergodic, what is the proportion of people who are told that the President will run? Interpret the fact that this proportion is independent of the starting state.
Show that Example 11.8 is an ergodic chain, but not a regular chain. Show that its fixed probability vector w is a binomial distribution.
Toss a fair die repeatedly. Let Sn denote the total of the outcomes through the nth toss. Show that there is a limiting value for the proportion of the first n values of Sn that are divisible by 7,
Prove that, in an r-state ergodic chain, it is possible to go from any state to any other state in at most r − 1 steps.
Prove that if P is the transition matrix of an ergodic chain, then (1/2) (I+P) is the transition matrix of a regular chain. Hint: Use Exercise 26.
(Alternate proof of Theorem 11.8) Let P be the transition matrix of an ergodic Markov chain. Let x be any column vector such that Px = x. Let M be the maximum value of the components of x. Assume
Consider the Markov chain with transition matrixFind the fundamental matrix Z for this chain. Compute the mean first passage matrix using Z.
Show that, for an ergodic Markov chain (see Theorem 11.16),The second expression above shows that the number K is independent of i. The number K is called Kemeny€™s constant. A prize was
A highly simplified game of “Monopoly” is played on a board with four squares as shown in Figure 11.8. You start at GO. You roll a die and move clockwise around the board a number of squares
Assume that an ergodic Markov chain has states s1, s2, . . . , sk. Let S(n) j denote the number of times that the chain is in state sj in the first n steps. Let w denote the fixed probability row
A rat runs through the maze shown in Figure 11.7. At each step it leaves the room it is in by choosing at random one of the doors out of the room. (a) Give the transition matrix P for this Markov
Consider a random walk on a circle of circumference n. The walker takes one unit step clockwise with probability p and one unit counterclockwise with probability q = 1 − p. Modify the program
Let P be the transition matrix of an ergodic Markov chain and P* the reverse transition matrix. Show that they have the same fixed probability vector w.
Show that any ergodic Markov chain with a symmetric transition matrix (i.e., pij = pji) is reversible.
An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector w). The mean time until the next occurrence of state si is m¯i = Σk wkmki + wiri. Show that m¯i =
A perpetual craps game goes on at Charley’s. Jones comes into Charley’s on an evening when there have already been 100 plays. He plans to play until the next time that snake eyes (a pair of ones)
Let X be a random variable which can take on countably many values. Show that X cannot be uniformly distributed.
Under the same conditions as in the preceding exercise, can you describe a procedure which, if used, would produce each possible outcome with the same probability? Can you describe such a procedure

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