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applied statistics and multivariate

Applied Statistics And Probability For Engineers 5th Edition Douglas C. Montgomery, George C. Runger - Solutions

It is possible to construct a nonparametric tol- erance interval that is based on the extreme values in a random sample of size n from any continuous population. If p is the minimum proportion of the population con- tained between the smallest and largest sample observa- tions with confidence 1-a,
Consider a two-sided confidence interval for the mean when or is known: wherea, a, a.lfa,a, a/2, we have the usual 100(-a)% confidence interval for u. In the above, whena, a, the interval is not symmetric about The length of the interval is La(+)/V. Prove that the length of the interval is
An electrical component has a time-to-failure (or lifetime) distribution that is exponential with param- eter , so the mean lifetime is = 1/A. Suppose that a sample of n of these components is put on test, and let X, be the observed lifetime of component. The test con- tinues only until the rth
An article in the Journal of Human Nutrition and Dietetics "The Validation of Energy and Protein Intakes by Doubly Labeled Water and 24-Hour Urinary Nitrogen Excretion in Post-Obese Subjects" (1995, Vol. 8, pp. 51-64)] showed the energy intake expressed as a basal metabolic rate, BMR (MJ) 5.40 5.67
An article in the Journal of Applied Physiology ["Humidity Does Not Affect Central Nervous System Oxygen Toxicity" (2001, Vol. 91, pp. 1327-1333)] reported that central nervous system (CNS) oxygen toxicity can appear in humans on exposure to oxygen pressures >180 kPa. CNS oxygen toxi- city can
An article in Engineering Horizons (Spring 1990, p. 26) reported that 117 of 484 new engineering graduates were planning to continue studying for an advanced degree. Consider this as a random sample of the 1990 graduating class. (a) Find a 90% confidence interval on the proportion of such graduates
The tar content in 30 samples of cigar tobacco follows: 1.542 1.585 1.532 1.466 1.499 1.611 1.622 1.440 1.608 1.520 1.466 1.546 1.494 1.548 1.626 1.478 1.542 1.511 1.459 1.533 1.532 1.523 1.397 1.487 1.598 1.498 1.600 1.504 1.545 1.558 (a) Is there evidence to support the assumption that the tar
Consider the baseball coefficient of restitution data in Exercise 8-92.Suppose that any baseball that has a coefficient of restitution that exceeds 0.635 is considered too lively. Based on the available data, what proportion of the baseballs in the sampled population are too lively? Find a 95%
During the 1999 and 2000 baseball seasons, there was much speculation that the unusually large number of home runs that were hit was due at least in part to a livelier ball. One way to test the "liveliness" of a baseball is to launch the ball at a vertical surface with a known velocity, and measure
The maker of a shampoo knows that customers like this product to have a lot of foam. Ten sample bottles of the product are selected at random and the foam heights observed are as follows (in millimeters): 210, 215, 194, 195, 211, 201, 198, 204, 208, and 196.(a) Is there evidence to support the
Consider the compressive strength of concrete data from Exercise 8-87.Find a 95% prediction interval on the next sample that will be tested.
Consider the hemoglobin data in Exercise 8-86.Find the following: (a) An interval that contains 95% of the hemoglobin values with 90% confidence. (b) An interval that contains 99% of the hemoglobin values with 90% confidence.
An operating system for a personal computer has been studied extensively, and it is known that the standard deviation of the response time following a particular command is = 8 milliseconds. A new version of the operating system is installed, and we wish to estimate the mean response time for the
The article "Mix Design for Optimal Strength Development of Fly Ash Concrete" (Cement and Concrete Research, 1989, Vol. 19, No. 4, pp. 634-640) investigates the compressive strength of concrete when mixed with fly ash (a mixture of silica, alumina, iron, magnesium oxide, and other ingredients). The
An article in the Journal of Sports Science (1987, Vol. 5.pp. 261-271) presents the results of an investigation of the hemoglobin level of Canadian Olympic ice hockey players. The data reported are as follows (in g/dl): 15.3 16.0 14.4 16.2 16.2 14.9 15.7 15.3 14.6 15.7 16.0 15.0 15.7 16.2 14.7 14.8
A normal population has known mean = 50 and variance o = 5.What is the approximate probability that the sample variance is greater than or equal to 7.44? less than or equal to 2.56? For a random sample of size (a) (b) (c) 16 30 71 (d) Compare your answers to parts (a)-(e) for the approxi- mate
A normal population has a known mean of 50 and unknown variance. (a) A random sample of n = 16 is selected from this popula tion, and the sample results are 52 and s = 8.How unusual are these results? That is, what is the probability of observing a sample average as large as 52 (or larger) if the
Consider the confidence interval for with known standard deviation : wherea, a,a. Let a 0.05 and find the interval for aaa/2=0.025. Now find the interval for the case a=0.01 and 0.04.Which interval is shorter? Is there any advantage to a "symmetric" confidence interval?
Consider the bottle-wall thickness measurements described in Exercise 8-40.(a) Compute a 90% tolerance interval on bottle-wall thick- ness that has confidence level 90%. (b) Compute a 90% lower tolerance bound on bottle-wall thickness that has confidence level 90%. Why would a lower tolerance bound
Consider the fuel rod enrichment data described in Exercise 8-41.Compute a 99% tolerance interval on rod enrichment that has confidence level 95%. Compare the length of the tolerance interval with the length of the 95% CI on the population mean.
Consider the strength-of-concrete data in Exercise 8-37.Compute a 90% tolerance interval on the compressive strength of the concrete that has 90% confidence.
Consider the television tube brightness data in Exercise 8-35.Compute a 99% tolerance interval on the brightness of the television tubes that has confidence level 95%. Compare the length of the tolerance interval with the length of the 99% Cl on the population mean. Which interval is shorter?
Consider the suspension rod diameter data in Exercise 8-38.Compute a 95% tolerance interval on the diameter of the rods described that has 90% confidence. Compare the length of the tolerance interval with the length of the 95% CI on the population mean. Which interval is shorter? Discuss the dif-
Consider the rainfall data in Exercise 8-33.Compute a 95% tolerance interval that has confidence level 95%. Compare the length of the tolerance interval with the length of the 95% CI on the population mean. Discuss the difference in interpretation of these two intervals.
Consider the margarine test described in Exercise 8-36.Compute a 99% tolerance interval on the polyunsaturated fatty acid in this particular type of margarine that has confi- dence level 95%. Compare the length of the tolerance inter- val with the length of the 99% CI on the population mean. Which
Consider the syrup-volume data in Exercise 8-29.Compute a 95% tolerance interval on the syrup volume that has confidence level 90% Compare the length of the tolerance interval with the length of the 95% CI on the population mean.
Consider the Izod impact test described in Exercise 8-28.Compute a 99% tolerance interval on the impact strength of PVC pipe that has confidence level 90%. Compare the length of the tolerance interval with the length of the 99% Cl on the population mean. Which interval is shorter? Discuss the
Consider the tire-testing data in Exercise 8-27.Compute a 95% tolerance interval on the life of the tires that has confidence level 95%. Compare the length of the toler- ance interval with the length of the 95% Cl on the population mean. Which interval is shorter? Discuss the difference in
How would you obtain a one-sided prediction bound on a future observation? Apply this procedure to obtain a 95% one-sided prediction bound on the wall thickness of the next bottle for the situation described in Exercise 8.40.
Consider the fuel rod enrichment data described in Exercise 8-41.Compute a 90% prediction interval on the enrichment of the next rod tested. Compare the length of the prediction interval with the length of the 99% CI on the population mean.
Consider the bottle-wall thickness measurements described in Exercise 8-40.Compute a 90% prediction interval on the wall thickness of the next bottle tested.
Consider the test on the compressive strength of con crete described in Exercise 8-37.Compute a 90% prediction interval on the next specimen of concrete tested.
Consider the suspension rod diameter measurements described in Exercise 8-38.Compute a 95% prediction inter- val on the diameter of the next rod tested. Compare the length of the prediction interval with the length of the 95% CI on the population mean
Consider the television tube brightness test described in Exercise 8-35.Compute a 99% prediction interval on the brightness of the next tube tested. Compare the length of the prediction interval with the length of the 99% CI on the popu lation mean
Consider the margarine test described in Exercise 8-36.Compute a 99% prediction interval on the polyunsaturated fatty acid in the next package of margarine that is tested. Compare the length of the prediction interval with the length of the 99% Cl on the population mean
Consider the rainfall in Exercise 8-33.Compute a 95% prediction interval on the rainfall for the next year. Compare the length of the prediction interval with the length of the 95% CI on the population mean
Consider the natural frequency of beams described in Exercise 8-32.Compute a 90% prediction interval on the diameter of the natural frequency of the next beam of this type that will be tested. Compare the length of the prediction interval with the length of the 90% Cl on the population mean.
Consider the syrup-dispensing measurements de- scribed in Exercise 8-29.Compute a 95% prediction interval on the syrup volume in the next beverage dispensed. Compare the length of the prediction interval with the length of the 95% CI on the population mean
Consider the Izod impact test described in Exercise 8-28.Compute a 99% prediction interval on the impact strength of the next specimen of PVC pipe tested. Compare the length of the prediction interval with the length of the 99% CI on the population mean.
Consider the tire-testing data described in Exercise 8-27.Compute a 95% prediction interval on the life of the next tire of this type tested under conditions that are similar to those employed in the original test. Compare the length of the prediction interval with the length of the 95% CI on the
An article in the Journal of the American Statistical Association (1990, Vol. 85, pp. 972-985) measured the weight of 30 rats under experiment controls. Suppose that there are 12 underweight rats. (a) Calculate a 95% two-sided confidence interval on the true proportion of rats that would show
Of 1000 randomly selected cases of lung cancer, 823 resulted in death within 10 years. (a) Calculate a 95% two-sided confidence interval on the death rate from lung cancer. (b) Using the point estimate of p obtained from the prelimi- nary sample, what sample size is needed to be 95% condi- dent
The 2004 presidential election exit polls from the crit- ical state of Ohio provided the following results. There were 2020 respondents in the exit polls and 768 were college grad- uates. Of the college graduates, 412 voted for George Bush. (a) Calculate a 95% confidence interval for the proportion
The fraction of defective integrated circuits produced in a photolithography process is being studied. A random sam- ple of 300 circuits is tested, revealing 13 defectives. (a) Calculate a 95% two-sided Cl on the fraction of defective circuits produced by this particular tool. (b) Calculate a 95%
An article in the Australian Journal of Agricultural Research ["Non-Starch Polysaccharides and Broiler Performance on Diets Containing Soyabean Meal as the Sole Protein Concentrate" (1993, Vol. 44, No. 8, pp. 1483-1499)] determined that the essential amino acid (Lysine) composition level of soybean
An article in Cancer Research ["Analyses of Litter- Matched Time-to-Response Data, with Modifications for Recovery of Interlitter Information" (1977, Vol. 37, pp. 3863-3868)] tested the tumorigenesis of a drug. Rats were randomly selected from litters and given the drug. The times of tumor
An article in Urban Ecosystems. "Urbanization and Warming of Phoenix (Arizona, USA): Impacts, Feedbacks and Mitigation" (2002, Vol. 6, pp. 183-203), mentions that Phoenix is ideal to study the effects of an urban heat island because it has grown from a population of 300,000 to approximately 3
The percentage of titanium in an alloy used in aero- space castings is measured in 51 randomly selected parts. The sample standard deviation is s = 0.37.Construct a 95% two- sided confidence interval for dr.
The sugar content of the syrup in canned peaches is normally distributed. A random sample of n = 10 cans yields a sample standard deviation of s = 4.8 milligrams. Calculate a 95% two-sided confidence interval for
Consider the situation in Exercise 8-44.Find a 99% lower confidence bound on the standard deviation.
A rivet is to be inserted into a hole. A random sample of 15 parts is selected, and the hole diameter is measured. The sample standard deviation of the hole diameter measure- ments is s = 0.008 millimeters. Construct a 99% lower confi- dence bound for or.
Determine the x2 percentile that is required to construct each of the following Cls: (a) Confidence level one-sided (upper) 95%, degrees of freedom = 24, (b) Confidence level = 99%, degrees of freedom = 9, one- sided (lower) (c) Confidence level = 90%, degrees of freedom = 19, two- sided.
An article in Nuclear Engineering International (February 1988, p. 33) describes several characteristics of fuel rods used in a reactor owned by an electric utility in Norway. Measurements on the percentage of enrichment of 12 rods were reported as follows: 2.94 2.90 3.00 2.90 2.75 3.00 2.95 2.75
The wall thickness of 25 glass 2-liter bottles was mea- sured by a quality-control engineer. The sample mean was = 4.05 millimeters, and the sample standard deviation was s-0.08 millimeter. Find a 95% lower confidence bound for mean wall thickness. Interpret the interval you have obtained.
An article in Computers & Electrical Engineering ["Parallel Simulation of Cellular Neural Networks" (1996, Vol. 22, pp. 61-84)] considered the speed-up of cellular neural networks (CNN) for a parallel general-purpose computing architecture based on six transputers in different areas. The data
A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected and the diameter is measured. The resulting data (in millime ters) are as follows: 8.24 8.25 8.20 8.23 8.24 8.21 8.26 8.26 8.20 8.25 8.23 8.23 8.19 8.28 8.24 (4) Check the assumption of
The compressive strength of concrete is being tested by a civil engineer. He tests 12 specimens and obtains the following data. 2216 2237 2249 2204 2225 2301 2281 2263 2318 2255 2275 2295 (a) Check the assumption that compressive strength is normally distributed. Include a graphical display in your
A particular brand of diet margarine was analyzed to determine the level of polyunsaturated fatty acid (in percent- ages). A sample of six packages resulted in the following data: 16.8, 17.2, 17.4, 16.9, 16.5, 17.1.(a) Check the assumption that the level of polyunsaturated fatty acid is normally
The brightness of a television picture tube can be eval- uated by measuring the amount of current required to achieve a particular brightness level. A sample of 10 tubes results in -317.2 and 15.7.Find (in microamps) a 99% confi dence interval on mean current required. State any necessary
The solar energy consumed (in trillion BTU) in the US. by year from 1989 to 2004 (source: US. Department of Energy Web site, http://www.eia.doc.gov/emeu) is shown in the table below. Read down, then right for year. 55.291 66.458 70.237 65.454 59.718 68.548 69.787 64.391 62.688 69.857 68.793 63.62
The Bureau of Meteorology of the Australian Government provided the mean annual rainfall (in milli- meters) in Australia 1983-2002 as follows (http://www.bom.go au climate change/rain03.txt): 499.2, 555.2, 398.8, 391.9, 453.4, 459.8, 483.7, 417.6, 469.2 452.4, 499.3, 340.6, 522.8, 469.9, 527.2,
An article in the Journal of Composite Materials (December 1989, Vol. 23, p. 1200) describes the effect of delam- ination on the natural frequency of beams made from composite laminates. Five such delaminated beams were subjected to loads, and the resulting frequencies were as follows (in hertz):
An article in Obesity Research Impaired Pressure Natriuresis in Obese Youths" (2003, Vol. 11, pp. 745-751)] de- scribed a study in which all meals were provided for 14 lean boys for three days followed by one stress (with a video-game task). The average systolic blood pressure (SBP) during the test
A postmix beverage machine is adjusted to release a certain amount of syrup into a chamber where it is mixed with carbonated water. A random sample of 25 beverages was found to have a mean syrup content of 1.10 fluid ounce and a standard deviation ofs 0.015 fluid ounce. Find a 95% Cl on the mean
An Izod impact test was performed on 20 specimens of PVC pipe. The sample mean is 7 = 1.25 and the sample stan dard deviation is s = 0.25.Find a 99% lower confidence bound on Izod impact strength.
A random sample has been taken from a normal dis- tribution. Output from a software package is given below: Variable N Mein SE Mean SDev x 1.58 (a) Fill in the missing quantities. Variance Sum 6.11 751.40 (b) Find a 95% CI on the population mean
A random sample has been taken from a normal dis- tribution. Output from a software package is given below: Variable N Mean SE Mean SDev Variance Sum 10 ? 0.507 1.605 7 251.848 (a) Fill in the missing quantities. (b) Find a 95% CI on the population mean.
Determine the f-percentile that is required to construct each of the following one-sided confidence intervals: (a) Confidence level = 95%, degrees of freedom = 14 (b) Confidence level = 99%, degrees of freedom = 19 (c) Confidence level = 99.9%, degrees of freedom = 24
Determine the e-pereemile that is required to construct each of the following two-sided confidence intervals: (a) Confidence level = 95%, degrees of freedom = 12 (b) Confidence level = 95%, degrees of freedom = 24 (e) Confidence level = 99%, degrees of freedom = 13 (d) Confidence level = 99.9%,
Find the values of the following percentiles: 15+ 532, and f.
An article in the Journal of Agricultural Science "The Use of Residual Maximum Likelihood to Model Grain Quality Characteristics of Wheat with Variety, Climatic and Nitrogen Fertilizer Effects" (1997, Vol. 128, pp. 135-142)] investigated means of wheat grain crude protein content (CP) and Hagberg
If the sample size is doubled, by how much is the length of the Cl on a in Equation 8-5 reduced? What happens to the length of the interval if the sample size is increased by a factor of four?
By how much must the sample size n be increased if the length of the Cl on a in Equation 8-5 is to be halved?
Suppose that in Exercise 8-14 we wanted the error in estimating the mean life from the two-sided confidence inter- val to be five hours at 95% confidence. What sample size should be used! 817.Suppose that in Exercise 8-14 we wanted the total width of the two-sided confidence interval on mean life
A civil engineer is analyzing the compressive strength of concrete. Compressive strength is normally distributed with = 1000(psi). A random sample of 12 specimens has a mean compressive strength of 7 = 3250 psi.(a) Construct a 95% two-sided confidence interval on mean compressive strength. (b)
The life in hours of a 75-watt light bulb is known to be normally distributed with a = 25 hours. A random sample of 20 bulbs has a mean life of T = 1014 hours. (a) Construct a 95% two-sided confidence interval on the mean life. (b) Construct a 95% lower-confidence bound on the mean life. Compare
A manufacturer produces piston rings for an auto- mobile engine. It is known that ring diameter is normally dis- tributed with a = 0.001 millimeters. A random sample of 15 rings has a mean diameter of -74.036 millimeters. (a) Construct a 99% two-sided confidence interval on the mean piston ring
The diameter of holes for a cable harness is known to have a normal distribution with r = 0.01 inch. A random sample of size 10 yields an average diameter of 1.5045 inch. Find a 99% two-sided confidence interval on the mean hole diameter
The yield of a chemical process is being studied. From previous experience, yield is known to be normally distributed and or 3.The past five days of plant operation have resulted in the following percent yields: 916, 88.73, 90.8, 89.95, and 91.3.Find a 95% two-sided confidence interval on the true
Past experience has indicated that the breaking strength of yarn used in manufacturing drapery material is normally distributed and that = 2 psi. A random sample of nine specimens is tested, and the average breaking strength is found to be 98 psi. Find a 95% two-sided confidence interval on the
Suppose that a = 100 random samples of water from a freshwater lake were taken and the calcium concentration (milligrams per liner) measured. A 95% CI on the mean cal- cium concentration is 0.49 p = 0.82.(a) Would a 99% CI calculated from the same sample data be longer or shotter?(b) Consider the
Following are two confidence interval estimates of the mean of the cycles to failure of an automotive door latch mechanism (the test was conducted at an elevated stress level to accelerate the failure). 312493215.7 3110.53230.1 (a) What is the value of the sample mean cycles to failure? (b) The
Consider the gain estimation problem in Exercise 8-4.(a) How large must be if the length of the 95% CI is to be 40? (b) How large must be if the length of the 99% Cl is to be 40?
A random sample has been taken from a normal distri- bution and the following confidence intervals constructed us- ing the same data: (37.53, 49.87) and (35.59, 51.81) (a) What is the value of the sample mean? (b) One of these intervals is a 99% CI and the other is a 95% Cl. Which one is the 95% CI
A random sample has been taken from a normal distri- bution and the following confidence intervals constructed us. ing the same data (38.02, 61.98) and (39.95, 60.05) (a) What is the value of the sample mean?(b) One of these intervals is a 95% CI and the other is a 90% CL. Which one is the 95% CI
A confidence interval estimate is desired for the gain in a circuit on a semiconductor device. Assume that gain is not- mally distributed with standard deviation or = 20.(a) Find a 95% CI for a when a 10 and 1000. (b) Find a 95% CI for when a 25 and 7 = 1000. (c) Find a 99% CI for when a = 10 and T
Consider the in Equation 8-5 gives 98% confidence? in Equation 8-5 gives 80% confidence! in Equation 8-5 gives 75% confidence! one-sided confidence interval expressions for a mean of a normal population. (a) What value of, would result in a 90% CT? (b) What value of, would result in a 95% CI? (c)
For a normal population with known variance a: (a) What value of (b) What value of (c) What value of
For a normal population with known variance o. answer the following questions: (a) What is the confidence level for the interval - 2.140/V +2140/Vi (b) What is the confidence level for the interval - 2.49/Va +249a/V (c) What is the confidence level for the interval - 1.850/Vi +1.85a/Vi
Explain the three types of interval estimates: confidence intervals, prediction intervals, and tolerance intervals
Construct a tolerance interval for a normal population
Construct prediction intervals for a future observation
Use a general method for constructing an approximate confidence interval on a parameter
Construct confidence intervals on a population proportion
Construct confidence intervals on the variance and standard deviation of a normal distribution
Construct confidence intervals on the mean of a normal distribution, using either the normal distribution or the t distribution method
for the distribution of the minimum of a sample from an exponential distribution with parameter A.] (b) It can be shown that (T/r) = 1/(r). How does this compare to ) in the uncensored experiment?
Censored Data. A common problem in indus try is life testing of components and systems. In this problem, we will assume that lifetime has an exponen- tial distribution with parameter A, so j = 1/4 = T is an unbiased estimate of p. When a components are tested until failure and the data X. X. X,
When the population has a normal distribution, the estimator median (X,- Fl. IX-XI. ---IX-Xl)/0.6745 is sometimes used to estimate the population standard deviation. This estimator is more robust to outliers than the usual sample standard deviation and usually does not dif- fer much from S when
Let X be a random variable with mean ja and variance o, and let X, XX, be a random sample of size n from X. Show that the statistic = (XX) is an unbiased estimator for or for an ap propriate choice for the constant k. Find this value for k

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