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Statistics For Engineers And Scientists 4th Edition William Navidi - Solutions
A new production process is being contemplated for the manufacture of stainless steel bearings. Measurements of the diameters of random samples of bearingsfrom the old and the new processes produced the following data: a. Can you conclude at the 5% level that one process yields a different mean
A shipment of fibers is not acceptable if the mean breaking strength of the fibers is less than 50 N. A large sample of fibers from this shipment was tested, and a 98% lower confidence bound for the mean breaking strength was computed to be 50.1 N. Someone suggests using these data to test the
The following MINITAB output presents the results of a hypothesis test for the difference μX?? μYbetween two population means. a. Is this a one-tailed or two-tailed test? b. What is the null hypothesis? c. Can H0 be rejected at the 1% level? How can you tell? Two -sample I for X vs Y N 10 10
Consider the following two samples: a. Show that both samples have the same mean and variance. b. Use the Wilcoxon rank-sum test to test the null hypothesis that the population means are equal. What do you conclude? c. Do the assumptions of the rank-sum test appear to be satisfied? Explain why or
A scientist computes a 90% confidence interval to be (4.38, 6.02). Using the same data, she also computes a 95% confidence interval to be (4.22, 6.18), and a 99% confidence interval to be (3.91, 6.49). Now she wants to test H0 :μ = 4 versus H1 :μ ≠ 4. Regarding the P-value, which one of the
Refer to Exercise 12 in Section 5.6. Can you conclude that the mean permeability coefficient at 60?C differs from that at 61?C? Refer to Exercise 12 60°C: 54 51 61 67 57 69 60 60 63 62 61°C: 58 60 66 66 68 61 60
Refer to Exercise 14. For which null hypothesis will P = 0.05?i. H0 :μ = 1.2ii. H0 :μ ≤ 1.2iii. H0 :μ ≥ 1.2Refer to Exercise 14. Fill in the blank: A 95% confidence interval for μ is (1.2, 2.0). Based on the data from which the confidence interval was constructed, someone wants to test
The following MINITAB output presents the results of a hypothesis test for a population mean μ. a. Is this a one-tailed or two-tailed test? b. What is the null hypothesis? c. What is the P-value? d. Use the output and an appropriate table to compute the P-value for the test of H0 :μ ?? 73.6
The following MINITAB output presents the results of a hypothesis test for the difference p1?? p2between two population proportions. a. Is this a one-tailed or two-tailed test? b. What is the null hypothesis? c. Can H0 be rejected at the 5% level? How can you tell? Test and CI for Two Proportions
Refer to Exercise 11 in Section 5.6. Can you conclude that the mean sodium content is higher for brand B than for brand A? Refer to Exercise 11 Brand A 34.36 31.26 37.36 28.52 33.14 32.74 34.34 34.33 30.95 Brand B 41.08 38.22 39.59 38.82 36.24 37.73 35.03 39.22 34.13 34.33 34.98 29.64 40.60
Electric motors are assembled on four different production lines. Random samples of motors are taken from each line and inspected. The number that pass and that fail the inspection are counted for each line, with the following results: Can you conclude that the failure rates differ among the four
In a study of the relationship between the Brinell hardness (x) and tensile strength in ksi (y) of specimens of cold drawn copper, the least-squares line was y = −196.32 + 2.42x.a. Predict the tensile strength of a specimen whose Brinell hardness is 102.7.b. If two specimens differ in their
The processing of raw coal involves €œwashing,€ in which coal ash (nonorganic, incombustible material) is removed. The article €œQuantifying Sampling Precision for Coal Ash Using Gy€™s Discrete Model of the Fundamental Error€ (Journal of Coal
For each of the following data sets, explain why the correlation coefficient is the same as for the data set in Exercise 1.a.b. c. 3 5 6 4 х 4 10 y 4. 3. 11 21 31 41 51 61 61 71 х y 5 4 10 9.
In a test of military ordnance, a large number of bombs were dropped on a target from various heights. The initial velocity of the bombs in the direction of the ground was 0. Let y be the height in meters from which a bomb is dropped, let x be the time in seconds for the bomb to strike the ground,
A least-squares line is fit to a set of points. If the total sum of squares isand the error sum of squares is compute the coefficient of determination r2 E(yi – J)² = 9615 E(y; – ŷi)² = 1450
Eruptions of the Old Faithful geyser in Yellowstone National Park typically last from 1.5 to 5 minutes. Between eruptions are dormant periods, which typically last from 50 to 100 minutes. A dormant period can also be thought of as the waiting time between eruptions. The durations in minutes for 60
In an study similar to the one in Exercise 3, the relative humidity and ozone levels were measured for 120 days in another city. The MINITAB output follows. Assume that assumptions 1 through 4 on page 544 hold. a. What is the slope of the least-squares line? b. Find a 95% confidence interval for
A least-squares line is fit to a set of points. If the total sum of squares isand the error sum of squares iscompute the coefficient of determination r2. E(yi – y)² = 181.2 Yi)? = 33.9
The depth of wetting of a soil is the depth to which water content will increase owing to external factors. The article €œDiscussion of Method for Evaluation of Depth of Wetting in Residential Areas€ (J. Nelson, K. Chao, and D. Overton, Journal of Geotechnical and
True or false, and explain briefly:a. If the correlation coefficient is positive, then above-average values of one variable are associated with above-average values of the other.b. If the correlation coefficient is negative, then below-average values of one variable are associated with
Refer to Exercise 3.a. Plot the residuals versus the fitted values. Does the plot indicate any serious violations of the standard assumptions?b. Plot the residuals versus the order of the data. Does the plot indicate any serious violations of the standard assumptions?Refer to Exercise 3.Eruptions
Cardiologists use the short-range scaling exponent α1, which measures the randomness of heart rate patterns, as a tool to assess risk of heart attack. The article ??Applying Fractal Analysis to Short Sets of Heart Rate Variability Data?? (M. Pe˜na et al., Med Biol Eng Comput, 2009:709??717)
In a study relating the degree of warping, in mm, of a copper plate (y) to temperature in ?C (x), the foll owing summary statistics were calculated: n = 40, a. Compute the correlation r between the degree of warping and the temperature.? b. Compute the error sum of squares, the regression sum of
The article €œDrift in Posturography Systems Equipped with a Piezoelectric Force Platform: Analysis and Numerical Compensation€ (L. Quagliarella, N. Sasanelli, and V. Monaco, IEEE Transactions on Instrumentation and Measurement, 2008:997€“1004) reported the results of
In a study of ground motion caused by earthquakes, the peak velocity (in m/s) and peak acceleration (in m/s2) were recorded for five earthquakes. The results are presented in the following table.a. Compute the correlation coefficient between peak velocity and peak acceleration.b. Construct a
The article “Experimental Measurement of Radiative Heat Transfer in Gas-Solid Suspension Flow System†(G. Han, K. Tuzla, and J. Chen, AIChe Journal, 2002:1910–1916) discusses the calibration of a radiometer. Several measurements were made on the electromotive force readings of
The article €œOptimization of Medium Composition for Lipase Production by Candida rugosa NCIM 3462 Using Response Surface Methodology€ (A. Ragendran and V. Thangavelu, Can J. Microbiol, 2007:643€“655) describes a series of experiments in which lipase was produced from
The article ??Application of Radial Basis Function Neural Networks in Optimization of Hard Turning of AISI D2 Cold- Worked Tool Steel With a Ceramic Tool?? (S. Basak, U. Dixit, and J. Davim, Journal of Engineering Manufacture, 2007:987??998) presents the results of an experiment in which the
The following table presents shear strengths (in kN/mm) and weld diameters (in mm) for a sample of spot welds.a. Construct a scatterplot of strength (y) versus diameter (x). Verify that a linear model is appropriate.b. Compute the least-squares line for predicting strength from diameter.c. Compute
The article €œOxidation State and Activities of Chromium Oxides in CaO-SiO2-CrOxSlag System€ (Y. Xiao, L. Holappa, and M. Reuter, Metallurgical and Materials Transactions B, 2002:595€“603) presents the amount x (in mole percent) and activity coefficient y of CrO1.5for
Another chemical engineer is studying the same process as in Exercise 7, and uses the following experimental matrix.a. Compute the correlation between temperature and yield, between stirring rate and yield, and between temperature and stirring rate.b. Do these data provide good evidence that the
Structural engineers use wireless sensor networks to monitor the condition of dams and bridges. The article €œStatistical Analysis of Vibration Modes of a Suspension Bridge Using Spatially Dense Wireless Sensor Network€ (S. Pakzad and G. Fenves, Journal of Structural
In a sample of 300 steel rods, the correlation coefficient between diameter and length was r = 0.15. a. Find the P-value for testing H0 : ρ ≤ 0 vs. H1 :ρ > 0. Can you conclude that ρ > 0?b. Does the result in part (a) allow you to conclude that there is a strong correlation between
The article €œThe Role of Niche Breadth, Resource Availability and Range Position on the Life History of Butterflies€ (A. Komonen, A. Grapputo, et al., Oikos, 2004:41€“54) describes a study of several species of butterflies found in Finland. The following table
Three engineers are independently estimating the spring constant of a spring, using the linear model specified by Hooke’s law. Engineer A measures the length of the spring under loads of 0, 1, 3, 4, and 6 lb, for a total of five measurements. Engineer B uses the same loads, but repeats the
In the skin permeability example (Example 7.17) imagine that 95% confidence intervals are to be computed for the mean permeability for skin with resistances of 15, 20, and 25 k Ω. Which of the confidence intervals would be the shortest? Which would be the longest? Explain.
Phonics is an instructional method in which children are taught to connect sounds with letters or groups of letters. The article “Predictive Accuracy of Nonsense Word Fluency for English Language Learners” (M. Vanderwood, D. Linklater, and K. Healy, School Psychology Review 2008:5–17) reports
A materials scientist is experimenting with a new material with which to make beverage cans. She fills cans with liquid at room temperature, and then refrigerates them to see how fast they cool. According to Newton€™s law of cooling, if t is the time refrigerated and y is the temperature
Choose the best answer to fill in the blank. If an estimator is unbiased, theni. The estimator is equal to the true value.ii. The estimator is usually close to the true value.iii. The mean of the estimator is equal to the true value.iv. The mean of the estimator is usually close to the true value.
Each of three samples has been plotted on a normal probability plot. For each, say whether the sample appears to have come from an approximately normal population.(a)(b)(c) 0.999 0.99 0.95 0.9 0.75 0.5 0.25 0.1 0.05 0.01 0.001 0.999 0.99 0.95 0.9 0.75 0.5 0.25 0.1 0.05 0.01 0.001
Resistors are labeled 100 Ω. In fact, the actual resistances are uniformly distributed on the interval (95, 103).a. Find the mean resistance.b. Find the standard deviation of the resistances.c. Find the probability that the resistance is between 98 and 102 Ω.d. Suppose that resistances of
There are two competing designs for a certain semiconductor circuit. The lifetime of the first (in hours) is exponentially distributed with λ = 10−4, and the lifetime of the second is lognormally distributed with μ = 6 and σ2 = 5.4.a. Use a simulated sample of size 1000 to estimate the
Find the area under the normal curvea. To the left of z = 0.56.b. Between z = −2.93 and z = −2.06.c. Between z = −1.08 and z = 0.70.d. Outside z = 0.96 to z = 1.62.
Choose the best answer to fill in the blank. The variance of an estimator measuresi. how close the estimator is to the true value.ii. how close repeated values of the estimator are to each other.iii. how close the mean of the estimator is to the true value.iv. how close repeated values of the mean
The number of flaws in a given area of aluminum foil follows a Poisson distribution with a mean of 3 per m2. Let X represent the number of flaws in a 1 m2 sample of foil.a. P(X = 5)b. P(X = 0)c. P(X < 2)d. P(X > 1)e. μXf. σX
A certain brand of dinnerware set comes in three colors: red, white, and blue. Twenty percent of customers order the red set, 45% order the white, and 35% order the blue. Let X = 1 if a randomly chosen order is for a red set, let X = 0 otherwise; let Y = 1 if the order is for a white set, let Y = 0
Let X ∼ Bin(9, 0.4). Finda. P(X > 6)b. P(X ≥ 2)c. P(2 ≤ X < 5)d. P(2 < X ≤ 5)e. P(X = 0)f. P(X = 7)g. μXh. σ2X
The number of large cracks in a length of pavement along a certain street has a Poisson distribution with a mean of 1 crack per 100 m.a. What is the probability that there will be exactly 8 cracks in a 500 m length of pavement?b. What is the probability that there will be no cracks in a 100 m
As part of a quality-control study aimed at improving a production line, the weights (in ounces) of 50 bars of soap are measured. The results are as follows, sorted from smallest to largest.Construct a normal probability plot for these data. Do these data appear to come from an approximately normal
A 500-page book contains 250 sheets of paper. The thickness of the paper used to manufacture the book has mean 0.08 mm and standard deviation 0.01 mm.a. What is the probability that a randomly chosen book is more than 20.2 mm thick (not including the covers)?b. What is the 10th percentile of book
The article “Assessment of Dermopharmacokinetic Approach in the Bioequivalence Determination of Topical Tretinoin Gel Products” (L. Pershing, J. Nelson, et al., J Am Acad Dermatol 2003:740–751) reports that the amount of a certain antifungal ointment that is absorbed into the skin can be
The time between requests to a web server is exponentially distributed with mean 0.5 seconds.a. What is the value of the parameter λ?b. What is the median time between requests?c. What is the standard deviation?d. What is the 80th percentile?e. Find the probability that more than one second
Let X1 and X2 be independent, each with unknown mean μ and known variance σ2 = 1.a. Let μ̂1 = X1 + X2/2. Find the bias, variance, and mean squared error of μ̂1.b. Let μ̂2 = X1 + 2X2/3. Find the bias, variance, and mean squared error of μ̂2.c. Let μ̂3 = X1 + X2/4. Find the
Find the following probabilities:a. P(X = 2) when X ∼ Bin(4, 0.6)b. P(X > 2) when X ∼ Bin(8, 0.2)c. P(X ≤ 2) when X ∼ Bin(5, 0.4)d. P(3 ≤ X ≤ 5) when X ∼ Bin(6, 0.7)
A commuter encounters four traffic lights each day on her way to work. Let X represent the number of these that are red lights. The probability mass function of X is as follows.What is the probability that in a period of 100 days, the average number of red lights encountered is more than 2 per day?
The article “Stochastic Estimates of Exposure and Cancer Risk from Carbon Tetrachloride Released to the Air from the Rocky Flats Plant” (A. Rood, P. McGavran, et al., Risk Analysis, 2001:675–695) models the increase in the risk of cancer due to exposure to carbon tetrachloride as lognormal
The distance between flaws on a long cable is exponentially distributed with mean 12 m.a. Find the probability that the distance between two flaws is greater than 15 m.b. Find the probability that the distance between two flaws is between 8 and 20 m.c. Find the median distance.d. Find the standard
A traffic light at a certain intersection is green 50% of the time, yellow 10% of the time, and red 40% of the time. A car approaches this intersection once each day. Let X represent the number of days that pass up to and including the first time the car encounters a red light. Assume that each day
Let T ∼Γ(r, λ). If μT = 8 and σ2T = 16, find r and λ.
A cable is made up of four wires. The breaking strength of each wire is a normally distributed random variable with mean 10 kN and standard deviation 1 kN. The strength of the cable, using the brittle wire method, is estimated to be the strength of the weakest wire multiplied by the number of
If X ∼ N(2, 9), computea. P(X ≥ 2)b. P(1 ≤ X < 7)c. P(−2.5 ≤ X < −1)d. P(−3 ≤ X − 2 < 3)
Let X1, . . . , Xnbe a simple random sample from a N(μ, σ2) population. For any constant k > 0, defineConsider σ̂2k as an estimator of σ2.a. Compute the bias of σ̂2k in terms of k. The sample variance s2 is
Let X and Y be Bernoulli random variables. Let Z = X + Y.a. Show that if X and Y cannot both be equal to 1, then Z is a Bernoulli random variable.b. Show that if X and Y cannot both be equal to 1, then pZ = pX + pY.c. Show that if X and Y can both be equal to 1, then Z is not a Bernoulli random
At a certain airport, 75% of the flights arrive on time. A sample of 10 flights is studied.a. Find the probability that all 10 of the flights were on time.b. Find the probability that exactly eight of the flights were on time.c. Find the probability that eight or more of the flights were on time.
A simple random sample X1, . . . , Xn is drawn from a population, and the quantities ln X1, . . . , ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false:a. X1, . . . , Xn come from a population that is approximately lognormal.b. X1, . . . ,
A data center contains 1000 computer servers. Each server has probability 0.003 of failing on a given day.a. What is the probability that exactly two servers fail?b. What is the probability that fewer than 998 servers function?c. What is the mean number of servers that fail?d. What is the standard
Construct a normal probability plot for the PM data in Table 1.2. Do the PM data appear to come from a normal population?Table 1.2 7.59 2.06 8.86 8.67 5.23 12.26 4.35 3.01 23.38 6.28 9.24 13.63 6.95 3.22 5.66 9.14 6.07 5.54 3.46 2.44 13.02 6.04 1.85 4.04 4.40 9.52 17.11 3.57 19.91 3.84 8.50 2.37
Bags checked for a certain airline flight have a mean weight of 15 kg with a standard deviation of 5 kg. A random sample of 60 bags is drawn.a. What is the probability that the sample mean weight is less than 14 kg?b. Find the 70th percentile of the sample mean weights.c. How many bags must be
Someone claims that the waiting time, in minutes, between hits at a certain website has the exponential distribution with parameter λ = 1.a. Let X be the waiting time until the next hit. If the claim is true, what is P(X ≥ 5)?b. Based on the answer to part (a), if the claim is true, is five
Refer to Exercise 5. Suppose 10 circuits are constructed. Find the probability that 8 or more have voltages less than 200 volts.Refer to Exercise 5.If a resistor with resistance R ohms carries a current of I amperes, the potential difference across the resistor, in volts, is given by V = IR.
The lifetime, in years, of a type of small electric motor operating under adverse conditions is exponentially distributed with λ = 3.6. Whenever a motor fails, it is replaced with another of the same type. Find the probability that fewer than six motors fail within one year.
The temperature recorded by a certain thermometer when placed in boiling water (true temperature 100°C) is normally distributed with mean μ = 99.8°C and standard deviation 0.1°C.a. What is the probability that the thermometer reading is greater than 100°C?b. What is the probability that the
Let X1, . . . , Xn be a random sample from a population with the Poisson(λ) distribution. Find the MLE of λ.
One out of every 5000 individuals in a population carries a certain defective gene. A random sample of 1000 individuals is studied.a. What is the probability that exactly one of the sample individuals carries the gene?b. What is the probability that none of the sample individuals carries the
Two dice are rolled. Let X = 1 if the dice come up doubles and let X = 0 otherwise. Let Y = 1 if the sum is 6, and let Y = 0 otherwise. Let Z = 1 if the dice come up both doubles and with a sum of 6 (that is, double 3), and let Z = 0 otherwise.a. Let pX denote the success probability for X. Find
A fair die is rolled 8 times.a. What is the probability that the die comes up 6 exactly twice?b. What is the probability that the die comes up an odd number exactly five times?c. Find the mean number of times a 6 comes up.d. Find the mean number of times an odd number comes up.e. Find the standard
In the article “Occurrence and Distribution of Ammonium in Iowa Groundwater” (K. Schilling, Water Environment Research, 2002:177–186), ammonium concentrations (in mg/L) were measured at a large number of wells in the state of Iowa. The mean concentration was 0.71, the median was 0.22, and the
Construct a normal probability plot for the logs of the PM data in Table 1.2. Do the logs of the PMdata appear to come from a normal population?. Table 1.2 13.63 6.95 7.59 6.07 3.46 2.44 13.02 23.38 9.24 6.04 1.85 6.28 5.23 5.54 3.01 3.22 2.06 4.04 17.11 12.26 19.91 8.50 7.81 7.18 18.64 7.10 5.66
The amount of warpage in a type of wafer used in the manufacture of integrated circuits has mean 1.3 mm and standard deviation 0.1 mm. A random sample of 200 wafers is drawn.a. What is the probability that the sample mean warpage exceeds 1.305 mm?b. Find the 25th percentile of the sample mean.c.
If X ∼ Geom(p), what is the most probable value of X?i. 0ii. 1/piii. piv. 1v. (1 − p)/p2
Let T ∼ Weibull (0.5, 3).a. Find μT.b. Find σT.c. Find P(T < 1).d. Find P(T > 5).e. Find P(2 < T < 4).
Can the plot in Exercise 6 be used to determine whether the PM data appear to come from a lognormal population? Explain.
The time spent by a customer at a checkout counter has mean 4 minutes and standard deviation 2 minutes.a. What is the probability that the total time taken by a random sample of 50 customers is less than 180 minutes?b. Find the 30th percentile of the total time taken by 50 customers.
A radioactive mass emits particles according to a Poisson process at a mean rate of 2 per second. Let T be the waiting time, in seconds, between emissions.a. What is the mean waiting time?b. What is the median waiting time?c. Find P(T > 2).d. Find P(T < 0.1).e. Find P(0.3
Choose the best answer, and explain. If X is a random variable with a lognormal distribution, theni. The mean of X is always greater than the median.ii. The mean of X is always less than the median.iii. The mean may be greater than, less than, or equal to the median, depending on the value of σ.
A process that fills packages is stopped whenever a package is detected whose weight falls outside the specification. Assume that each package has probability 0.01 of falling outside the specification and that the weights of the packages are independent.a. Find the mean number of packages that will
If T is a continuous random variable that is always positive (such as a waiting time), with probability density function f (t) and cumulative distribution function F(t), then the hazard function is defined to be the functionThe hazard function is the rate of failure per unit time, expressed as a
In order to increase the lifetime of the system, the engineers have a choice between replacing component A with one whose lifetime is distributed Exp(1/2), or replacing component B with one whose lifetime is distributed Exp(1/3).a. Generate, by simulation, a large number (at least 1000) of system
Weights of female cats of a certain breed are normally distributed with mean 4.1 kg and standard deviation 0.6 kg.a. What proportion of female cats have weights between 3.7 and 4.4 kg?b. A certain female cat has a weight that is 0.5 standard deviations above the mean. What proportion of female cats
The number of cars arriving at a given intersection follows a Poisson distribution with a mean rate of 4 per second.a. What is the probability that 3 cars arrive in a given second?b. What is the probability that 8 cars arrive in three seconds?c. What is the probability that more than 3 cars arrive
Let X1, . . . , Xn be a random sample from a N(μ, 1) population. Find the MLE of μ.
A general contracting firm experiences cost overruns on 20% of its contracts. In a company audit, 20 contracts are sampled at random.a. What is the probability that exactly four of them experience cost overruns?b. What is the probability that fewer than three of them experience cost overruns?c.
A certain type of plywood consists of five layers. The thicknesses of the layers are independent and normally distributed with mean 5 mm and standard deviation 0.2 mm.a. Find the mean thickness of the plywood.b. Find the standard deviation of the thickness of the plywood.c. Find the probability
In the article €œAssessment of Dermatopharmacokinetic Approach in the Bioequivalence Determination of Topical Tretinoin Gel Products€ (L. Pershing, J. Nelson, et al., Journal of the American Academy of Dermatology, 2003:740€“751), measurements of the concentration of
Drums labeled 30 L are filled with a solution from a large vat. The amount of solution put into each drum is random with mean 30.01 L and standard deviation 0.1 L.a. What is the probability that the total amount of solution contained in 50 drums is more than 1500 L?b. If the total amount of
The distance between consecutive flaws on a roll of sheet aluminum is exponentially distributed with mean distance 3 m. Let X be the distance, in meters, between flaws.a. What is the mean number of flaws per meter?b. What is the probability that a 5 m length of aluminum contains exactly two flaws?
A manufacturer claims that the tensile strength of a certain composite (in MPa) has the lognormal distribution with μ = 5 and σ = 0.5. Let X be the strength of a randomly sampled specimen of this composite.a. If the claim is true, what is P(X < 20)?b. Based on the answer to part (a), if the
A computer program has a bug that causes it to fail once in every thousand runs, on average. In an effort to find the bug, independent runs of the program will be made until the program has failed five times.a. What is the mean number of runs required?b. What is the standard deviation of the number
The lifetime of a certain battery is modeled with the Weibull distribution with α = 2 and β = 0.1.a. What proportion of batteries will last longer than 10 hours?b. What proportion of batteries will last less than 5 hours?c. What proportion of batteries will last longer than 20 hours?d. The hazard
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