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Questions and Answers of Statistics Engineers Scientists

A 95% confidence interval for a population mean is computed from a sample of size 400. Another 95% confidence interval will be computed from a sample of size 100 drawn from the same population.Choose
A meteorologist measures the temperature in downtown Denver at noon on each day for one year. The 365 readings average 57°F and have standard deviation 20°F. The meteorologist computes a 95%
Sixty-four independent measurements were made of the speed of light. They averaged 299,795 km/s and had a standard deviation of 8 km/s. True or false:a. A 95% confidence interval for the speed of
Based on a sample of repair records, an engineer calculates a 95% confidence interval for the mean cost to repair a fiber-optic component to be ($140, $160). A supervisor summarizes this result in a
The following are summary statistics for a data set. Would it be appropriate to use the Student€™s t distribution to construct a confidence interval from these data? Explain. N Mean Median
The article “Application of Surgical Navigation to Total Hip Arthroplasty” (T. Ecker and S. Murphy, Journal of Engineering in Medicine, 2007:699–712) reports that in a sample of 123 hip
A group of 78 people enrolled in a weight-loss program that involved adhering to a special diet and to a daily exercise program. After six months, their mean weight loss was 25 pounds, with a sample
The article "Automatic Filtering of Outliers in RR Intervals Before Analysis of Heart Rate Variability in Holter Recordings: a Comparison with Carefully Edited Data" (M. Karlsson, et al., Biomedical
True or false: The Student’s t distribution may be used to construct a confidence interval for the mean of any population, so long as the sample size is small.
Refer to Exercise 1. Find a 95% lower confidence bound for the proportion of automobiles whose emissions exceed the standard.Refer to Exercise 1In a simple random sample of 70 automobiles registered
The article “HIV-positive Smokers Considering Quitting: Differences by Race/Ethnicity” (E. Lloyd- Richardson, C. Stanton, et al., Am J Health Behav, 2008:3–15) surveyed 444 HIV-positive
The article “Hatching Distribution of Eggs Varying in Weight and Breeder Age” (S. Viera, J. Almeida, et al., Brazilian Journal of Poultry Science 2005: 73–78) presents the results of a study in
Find the value of tn−1,α needed to construct an upper or lower confidence bound in each of the situations in Exercise 1.Data from exercise 1Find the value of tn−1,α/2 needed to construct a
The article “Modeling Arterial Signal Optimization with Enhanced Cell Transmission Formulations” (Z. Li, Journal of Transportation Engineering 2011:445–454) presents a new method for timing
Find the levels of the confidence intervals that have the following values of zα/2:a. zα/2 = 1.96b. zα/2 = 2.17c. zα/2 = 1.28d. zα/2 = 3.28
During a recent drought, a water utility in a certain town sampled 100 residential water bills and found that 73 of the residences had reduced their water consumption over that of the previous
The article “Some Parameters of the Population Biology of Spotted Flounder (Ciutharus linguatula Linnaeus, 1758) in Edremit Bay (North Aegean Sea)” (D. T¨urker, B. Bayhan, et al., Turkish
A stress analysis was conducted on random samples of epoxy-bonded joints from two species of wood. A random sample of 120 joints from species A had a mean shear stress of 1250 psi and a standard
Refer to Exercise 4. Find a 99% lower confidence bound for the proportion of HIV-positive smokers who have used a nicotine patch.Refer to Exercise 4.The article “HIV-positive Smokers Considering
Refer to Exercise 8.a. Find a 99% upper confidence bound for the mean temperature.b. The claim is made that the mean temperature is less than 349.5◦F. With what level of confidence can this
Oven thermostats were tested by setting them to 350◦F and measuring the actual temperature of the oven. In a sample of 67 thermostats, the average temperature was 348.2◦F and the standard
Six measurements are taken of the thickness of a piece of 18-gauge sheet metal. The measurements (in mm) are: 1.316, 1.308, 1.321, 1.303, 1.311, and 1.310. a. Make a dotplot of the six values.b.
An electrical engineer wishes to compare the mean lifetimes of two types of transistors in an application involving high-temperature performance. Asample of 60 transistors of type A were tested and
In a sample of 60 electric motors, the average efficiency (in percent) was 85 and the standard deviation was 2.a. Find a 95% confidence interval for the mean efficiency.b. Find a 99.5% confidence
Fission tracks are trails found in uranium-bearing minerals, left by fragments released during fission events. The article “Yo-yo Tectonics of the Nigde Massif During Wrenching in Central
When the light turns yellow, should you stop or go through it? The article “Evaluation of Driver Behavior in Type II Dilemma Zones at High-Speed Signalized Intersections” (D. Hurwitz, M. Knodler,
Refer to Exercise 7.a. Find a 95% lower confidence bound for the mean capacity of this type of battery.b. An engineer claims that the mean capacity is greater than 175 ampere-hours. With what level
The sugar content in a one-cup serving of a certain breakfast cereal was measured for a sample of 140 servings. The average was 11.9 g and the standard deviation was 1.1 g.a. Find a 95% confidence
The article “The Prevalence of Daytime Napping and Its Relationship to Nighttime Sleep” (J. Pilcher, K. Michalkowski, and R. Carrigan), Behavioral Medicine, 2001:71–76) presents results of a
Refer to Exercise 5.a. Find a 95% lower confidence bound for the mean strength.b. Someone says that the mean strength is less than 50.4 kN. With what level of confidence can this statement be
Refer to Exercise 11. Ten more welds will be made in order to increase the precision of the confidence interval. Which would increase the precision the most, cooling all 10 welds at the rate of
In a random sample of 150 customers of a high-speed internet provider, 63 said that their service had been interrupted one or more times in the past month.a. Find a 95% confidence interval for the
Let Z ˆ¼ N(0, 1), and let X = Ïƒ Z + Î¼ where Î¼ and Ïƒ > 0 are constants. Let represent the cumulative distribution function of Z, and
Let X ˆ¼ Poisson(Î»).a. Show that if x is a positive integer, thenb. Show that if X ˆ¼ Poisson(Î»), the most probable value for X is the greatest
Let X ˆ¼ Bin(n, p).a. Show that if x is an integer between 1 and n inclusive, thenb. Show that if X ˆ¼ Bin(n, p), the most probable value for X is the greatest integer less
A stick of length 1 is broken at a point chosen uniformly along its length. One piece is used as the length of a rectangle, and the other is used as the width. Find the mean area of a rectangle
Refer to Example 4.14. Estimate the probability that exactly one of the four tires has a flaw, and find the uncertainty in the estimate.
Chebyshev’s inequality states that for any random variable X with mean μ and variance σ2, and for any positive number k, P(|X − μ| ≥ kσ) ≤ 1/k2. Let X ∼ N(μ, σ2). Compute P(|X −μ|
Radioactive mass 1 emits particles at a mean rate of Î»1per second, and radioactive mass 2 emits particles at a mean rate of Î»2per second. Mass 1 is selected with probability
There are 30 restaurants in a certain town. Assume that four of them have health code violations. A health inspector chooses 10 restaurants at random to visit.a. What is the probability that two of
The quality-assurance program for a certain adhesive formulation process involves measuring how well the adhesive sticks a piece of plastic to a glass surface. When the process is functioning
One design for a system requires the installation of two identical components. The system will work if at least one of the components works. An alternative design requires four of these components,
Refer to Exercise 23. Assume that if m = 0, the value s = −1.5 is sent, and if m = 1, the value s = 1.5 is sent. The value received is X, where X = s + E, and E ∼ N(0, 0.64). If X ≤ 0.5, then
An alternative to the lognormal distribution for modeling highly skewed populations is the Pareto distribution with parameters Î¸ and r. The probability density function isThe parameters
Let X ∼ Bin(n, p), and let Y = n − X. Show that Y ∼ Bin(n, 1 − p).
The molarity of a solute in solution is defined to be the number of moles of solute per liter of solution (1 mole = 6.02 × 1023 molecules). If X is the molarity of a solution of sodium chloride
Refer to Example 4.27. Estimate the probability that a 1 m2 sheet of aluminum has exactly one flaw, and find the uncertainty in this estimate. Example 4.27 The number of flaws on a sheet of
Two resistors, with resistances R1 and R2, are connected in series. R1 is normally distributed with mean 100 Ω and standard deviation 5 Ω, and R2 is normally distributed with mean 120 Ω and
Let X ∼ Bin(7, 0.3). Finda. P(X = 1)b. P(X = 2)c. P(X < 1)d. P(X > 4)e. μXf. σ2X
Refer to Exercise 14. Assume that l = 10.0 cm ± 0.5% and d = 10.4 cm ± 0.5%.a. Estimate R, and find the relative uncertainty in the estimate.Does the relative uncertainty dependonk?b. Assume that
The article “Effect of Varying Solids Concentration and Organic Loading on the Performance of Temperature Phased Anaerobic Digestion Process” (S. Vandenburgh and T. Ellis, Water Environment
A measurement of the circumference of a disk has an uncertainty of 1.5 mm. How many measurements must be made so that the diameter can be estimated with an uncertainty of only 0.5 mm?
Assume that X, Y, and Z are independent measurements, and that the relative uncertainty in X is 5%, the relative uncertainty in Y is 10%, and the relative uncertainty in Z is 15%. Find the relative
The boiling point of water is measured four times. The results are 110.01◦C, 110.02◦C, 109.99◦C, and 110.01◦C. Which of the following statements best describes this measuring process?i.
Assume that X, Y, and Z are independent measurements with X = 25 ± 1, Y = 5.0 ± 0.3, and Z = 3.5 ± 0.2. Find the uncertainties in each of the following quantities:a. XY + Zb. Z/(X + Y )c. √X( ln
Assume that X and Y are independent measurements with uncertainties σX = 0.3 and σY = 0.2. Find the uncertainties in the following quantities:a. 4Xb. X + 2Yc. 2X − 3Y
Two thermometers are calibrated by measuring the freezing point of glacial acetic acid, which is 16.6◦C. Equal numbers of measurements are taken with each thermometer. The result from the first
For some genetic mutations, it is thought that the frequency of the mutant gene in men increases linearly with age. If m1 is the frequency at age t1, and m2 is the frequency at age t2, then the
For some measuring processes, the uncertainty is approximately proportional to the value of the measurement. For example, a certain scale is said to have an uncertainty of ±2%. An object is weighed
An item is to be constructed by laying three components in a row. The length of each component will be measured.a. If the uncertainty in measuring the length of each component is 1.2 mm, what is the
The weight of an object is given as 67.2 ± 0.3 g. True or false:a. The weight was measured to be 67.2 g.b. The true weight of the object is 67.2 g.c. The bias in the measurement is 0.3 g.d. The
True or false: In any boxplot,a. The length of the whiskers is equal to 1.5 IQR, where IQR is the interquartile range.b. The length of the whiskers may be greater than 1.5 IQR, where IQR is the
Refer to Table 1.5 (in Section 1.3).a. Using the class intervals in the table, construct a histogram in which the heights of the rectangles are equal to the relative frequencies.b. Compare the
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 total
There are 10 employees in a particular division of a company. Their salaries have a mean of $70,000, a median of $55,000, and a standard deviation of $20,000. The largest number on the list is
Following are boxplots comparing the amount of econozole nitrate (in Î¼g/cm2) absorbed into skin for a brand name and a generic antifungal ointment (from the article €œImproved
In each of the following data sets, tell whether the outlier seems certain to be due to an error, or whether it could conceivably be correct.a. The length of a rod is measured five times. The
The article €œHydrogeochemical Characteristics of Groundwater in a Mid-Western Coastal Aquifer System€ (S. Jeen, J. Kim, et al., Geosciences Journal, 2001:339€“348)
Match each scatterplot to the statement that best describes it.(a)(b)(c)(d)i. The relationship between x and y is approximately linear.ii. The relationship between x and y is nonlinear.iii. There
For a list of positive numbers, is it possible for the standard deviation to be greater than the mean? If so, give an example. If not, explain why not.
Find a sample size for which the median will always equal one of the values in the sample.
A sample of 100 college students is selected from all students registered at a certain college, and it turns out that 38 of them participate in intramural sports. True or false:a. The proportion
Following are measurements of soil concentrations (in mg/kg) of chromium (Cr) and nickel (Ni) at 20 sites in the area of Cleveland, Ohio. These data are taken from the article €œVariation
A coin is tossed twice and comes up heads both times. Someone says, “There’s something wrong with this coin. A coin is supposed to come up heads only half the time, not every time.”a. Is it
Is the sample median always equal to one of the values in the sample? If so, explain why. If not, give an example.
True or false:a. A simple random sample is guaranteed to reflect exactly the population from which it was drawn.b. A simple random sample is free from any systematic tendency to differ from the
The specification for the pull strength of a wire that connects an integrated circuit to its frame is 10 g or more. Units made with aluminum wire have a defect rate of 10%. A redesigned manufacturing
If you wanted to estimate the mean height of all the students at a university, which one of the following sampling strategies would be best? Why? Note that none of the methods are true simple random
Refer to Exercise 1. The vendor begins using heavier packaging, which increases the weight of each package by 50 g.a. How does this affect the mean weight of the packages?b. How does this affect the
Forty-five specimens of a certain type of powder were analyzed for sulfur trioxide content. Following are the results, in percent. The list has been sorted into numerical order.a. Construct a
Each of the following processes involves sampling from a population. Define the population, and state whether it is tangible or conceptual.a. A chemical process is run 15 times, and the yield is
A vendor converts the weights on the packages she sends out from pounds to kilograms (1 kg ≈ 2.2 lb).a. How does this affect the mean weight of the packages?b. How does this affect the standard
True or false: For any list of numbers, half of them will be below the mean.
Refer to Exercise 4. Assume that T = 298.4 ± 0.2 K. Estimate V, and find the relative uncertainty in the estimate.Refer to Exercise 4.The velocity V of sound in air at temperature T is given by V =
The resistance R (in ohms) of a cylindrical conductor is given by R = kl/d2, where l is the length, d is the diameter, and k is a constant of proportionality. Assume that l = 14.0 ± 0.1 cm and d =
The mean yield from process A is estimated to be 80 ± 5, where the units are percent of a theoretical maximum. The mean yield from process B is estimated to be 90 ± 3. The relative increase
A student measures the spring constant k of a spring by loading it and measuring the extension. (According to Hooke’s law, if l is the load and e is the extension, then k = l/e.) Assume five
The conversion of cyclobutane (C4H8) to ethylene (C2H4) is a first-order reaction. This means that the concentration of cyclobutane at time t is given by ln C = ln C0 − kt, where C is the
An object is weighed four times, and the results, in milligrams, are 234, 236, 233, and 229. The object is then weighed four times on a different scale, and the results, in milligrams, are 236, 225,
Refer to Exercise 8. Assume the critical angle is measured to be 0.90 ± 0.01 rad. Estimate the refractive index and find the relative uncertainty in the estimate.Refer to Exercise 8The refractive

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