Questions and Answers of Statistics Engineers Scientists

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
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
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
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
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,
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
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
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
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
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
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
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
The processing of raw coal involves €œwashing,€ in which coal ash (nonorganic, incombustible material) is removed. The article €œQuantifying Sampling Precision for 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
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 =
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
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
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
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
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
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
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
The article €œDrift in Posturography Systems Equipped with a Piezoelectric Force Platform: Analysis and Numerical Compensation€ (L. Quagliarella, N. Sasanelli, and V. Monaco,
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.
The article â€œExperimental Measurement of Radiative Heat Transfer in Gas-Solid Suspension Flow Systemâ€ (G. Han, K. Tuzla, and J. Chen, AIChe Journal, 2002:1910â€“1916) discusses the
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.
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
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
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,
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
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
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
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,
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,
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
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
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.
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
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
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
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
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.
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
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
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
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 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
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)
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
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 +
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
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,
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
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
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
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(Î¼, Ïƒ2) population. For any constant k > 0, defineConsider ÏƒÌ‚2k as an estimator of
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,
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
In the article €œAssessment of Dermatopharmacokinetic Approach in the Bioequivalence Determination of Topical Tretinoin Gel Products€ (L. Pershing, J. Nelson, et al., Journal 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
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
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
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
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

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