1 Million+ Step-by-step solutions

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 8

The refractive index n of a piece of glass is related to the critical angle θ by n = 1/sin θ. Assume that the critical angle is measured to be 0.70 ± 0.02 rad. Estimate the refractive index, and find the uncertainty in the estimate.

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, 245, and 240. The average of all eight measurements will be used to estimate the weight. Someone suggests estimating the uncertainty in this estimate as follows: Compute the standard deviation of all eight measurements. Call this quantity s. The uncertainty is then s/√8. Is this correct? Explain.

The conversion of cyclobutane (C_{4}H_{8}) to ethylene (C_{2}H_{4}) is a first-order reaction. This means that the concentration of cyclobutane at time t is given by ln C = ln C_{0} − kt, where C is the concentration at time t, C_{0} is the initial concentration, t is the time since the reaction started, and k is the rate constant. Assume that C_{0} = 0.2 mol/L with negligible uncertainty. After 300 seconds at a constant temperature, the concentration is measured to be C = 0.174 ± 0.005 mol/L. Assume that time can be measured with negligible uncertainty.

a. Estimate the rate constant k, and find the uncertainty in the estimate. The units of k will be s^{−1}.

b. Find the relative uncertainty in k.

c. The half-life t_{1/2} of the reaction is the time it takes for the concentration to be reduced to one-half its initial value. The half-life is related to the rate constant by t_{1/2} = (ln 2)/k. Using the result found in part (a), find the uncertainty in the half-life.

d. Find the relative uncertainty in the half-life.

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 independent measurements are made, and the measured values of k (in N/m) are 36.4, 35.4, 38.6, 36.6, and 38.0.

a. Estimate the spring constant, and find the uncertainty in the estimate.

b. Find an approximate value for the uncertainty in the average of 10 measurements.

c. Approximately how many measurements must be made to reduce the uncertainty to 0.3 N/m?

d. A second spring, similar to the first, is measured once. The measured value for k is 39.3. Approximately how much is the uncertainty in this measurement?

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 obtained from process B is therefore estimated to be (90 − 80)/80 = 0.125. Find the uncertainty in this estimate.

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 = 4.4 ± 0.1 cm.

a. Estimate R, and find the uncertainty in the estimate. Your answer will be in terms of the proportionality constant k.

b. Which would provide the greater reduction in the uncertainty in R: reducing the uncertainty in l to 0.05 cm or reducing the uncertainty in d to 0.05 cm?

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 = 20.04√T , where T is measured in kelvins (K) and V is in m/s. Assume that T = 300 ± 0.4 K. Estimate V, and find the uncertainty in the estimate.

True or false: For any list of numbers, half of them will be below the mean.

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 deviation of the weights?

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 measured each time.

b. A pollster samples 1000 registered voters in a certain state and asks them which candidate they support for governor.

c. In a clinical trial to test a new drug that is designed to lower cholesterol, 100 people with high cholesterol levels are recruited to try the new drug.

d. Eight concrete specimens are constructed from a new formulation, and the compressive strength of each is measured.

e. A quality engineer needs to estimate the percentage of bolts manufactured on a certain day that meet a strength specification. At 3:00 in the afternoon he samples the last 100 bolts to be manufactured.

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 stem-and-leaf plot for these data.

b. Construct a histogram for these data.

c. Construct a dotplot for these data.

d. Construct a boxplot for these data. Does the boxplot show any outliers?

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 standard deviation of the weights?

Refer to Exercise 1.

A vendor converts the weights on the packages she sends out from pounds to kilograms (1 kg ≈ 2.2 lb).

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 samples.

i. Measure the heights of 50 students found in the gym during basketball intramurals.

ii. Measure the heights of all engineering majors.

iii. Measure the heights of the students selected by choosing the first name on each page of the campus phone book.

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 process, involving the use of gold wire, is being investigated. The goal is to reduce the rate of defects to 5% or less. Out of the first 100 units manufactured with gold wire, only 4 are defective. True or false:

a. Since only 4% of the 100 units were defective, we can conclude that the goal has been reached.

b. Although the sample percentage is under 5%, this may represent sampling variation, so the goal may not yet be reached.

c. There is no use in testing the new process, because no matter what the result is, it could just be due to sampling variation.

d. If we sample a large enough number of units, and if the percentage of defective units is far enough below 5%, then it is reasonable to conclude that the goal has been reached.

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 population from which it was drawn.

Is the sample median always equal to one of the values in the sample? If so, explain why. If not, give an example.

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 reasonable to conclude that something is wrong with the coin? Explain.

b. If the coin came up heads 100 times in a row, would it be reasonable to conclude that something is wrong with the coin? Explain.

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 â€œVariation in North American Regulatory Guidance for Heavy Metal Surface Soil Contamination at Commercial and Industrial Sitesâ€ (A. Jennings and J. Ma, J Environment Eng, 2007:587â€“609).

a. Construct a histogram for each set of concentrations.

b. Construct comparative boxplots for the two sets of concentrations.

c. Using the boxplots, what differences can be seen between the two sets of concentrations?

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 of students at this college who participate in intramural sports is 0.38.

b. The proportion of students at this college who participate in intramural sports is likely to be close to 0.38, but not equal to 0.38.

Find a sample size for which the median will always equal one of the values in the sample.

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.

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 isnâ€™t much of any relationship between x and y.

iv. The relationship between x and y is approximately linear, except for an outlier.

The article â€œHydrogeochemical Characteristics of Groundwater in a Mid-Western Coastal Aquifer Systemâ€ (S. Jeen, J. Kim, et al., Geosciences Journal, 2001:339â€“348) presents measurements of various properties of shallow groundwater in a certain aquifer system in Korea. Following are measurements of electrical conductivity (in microsiemens per centimeter) for 23 water samples.

a. Find the mean.

b. Find the standard deviation.

c. Find the median.

d. Construct a dotplot.

e. Find the 10% trimmed mean.

f. Find the first quartile.

g. Find the third quartile.

h. Find the interquartile range.

i. Construct a boxplot.

j. Which of the points, if any, are outliers?

k. If a histogram were constructed, would it be skewed to the left, skewed to the right, or approximately symmetric?

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 readings in centimeters are 48.5, 47.2, 4.91, 49.5, 46.3.

b. The prices of five cars on a dealer’s lot are $25,000, $30,000, $42,000, $110,000, $31,000. 1.3 Graphical

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 â€œImproved Bioequivalence Assessment of Topical Dermatological Drug Products Using Dermatopharmacokinetics.â€ B. Nâ€™Dri-Stempfer, W. Navidi, R. Guy, and A. Bunge, Pharmaceutical Research, 2009:316â€“328).

True or false:

a. The median amount absorbed for the brand name drug is greater than the 25th percentile of the amount absorbed for the generic drug.

b. The median amount absorbed for the brand name drug is greater than the median amount absorbed for the generic drug.

c. About half the sample values for the brand name drug are between 2 and 3.

d. There is a greater proportion of values outside the box for the brand name drug than for the generic drug.

e. Both samples are skewed to the right.

f. Both samples contain outliers.

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 $100,000. By accident, this number is changed to $1,000,000.

a. What is the value of the mean after the change?

b. What is the value of the median after the change?

c. What is the value of the standard deviation after the change?

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 of 349 alluvial wells in the state of Iowa. The mean concentration was 0.27, the median was 0.10, and the standard deviation was 0.40. If a histogram of these 349 measurements were drawn,

i. it would be skewed to the right.

ii. it would be skewed to the left.

iii. it would be approximately symmetric.

iv. its shape could not be determined without knowing the relative frequencies.

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 histogram in part (a) with the histogram in Figure 1.9, for which the heights are the densities. Are the shapes of the histograms the same?

c. Explain why the heights should not be set equal to the relative frequencies in this case.

d. Which classes are visually exaggerated by making the heights equal to the relative frequencies?

TABLE 1.5

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 interquartile range.

c. The length of the whiskers may be less than 1.5 IQR, where IQR is the interquartile range.

d. The values at the ends of the whiskers are always values in the data set used to construct the boxplot.

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 uncertainty in the measurement is 0.3 g.

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 uncertainty in the combined length of the three components?

b. If it is desired to estimate the length of the item with an uncertainty of 0.5 mm, what must be the uncertainty in the measurement of each individual component? Assume the uncertainties in the three measurements are the same.

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 on this scale.

a. Given that the reading is 100 g, express the uncertainty in this measurement in grams.

b. Given that the reading is 50 g, express the uncertainty in this measurement in grams.

For some genetic mutations, it is thought that the frequency of the mutant gene in men increases linearly with age. If m_{1} is the frequency at age t_{1}, and m_{2} is the frequency at age t_{2}, then the yearly rate of increase is estimated by r = (m_{2} − m_{1})/(t_{2} − t_{1}). In a polymerase chain reaction assay, the frequency in 20-year-old men was estimated to be 17.7 ± 1.7 per μg DNA, and the frequency in 40-year-old men was estimated to be 35.9 ± 5.8 per μg DNA. Assume that age is measured with negligible uncertainty.

a. Estimate the yearly rate of increase, and find the uncertainty in the estimate.

b. Find the relative uncertainty in the estimated rate of increase.

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 thermometer is 16.4 ± 0.2◦C and the result from the second thermometer is 16.8 ± 0.1◦C.

a. Is it possible to tell which thermometer is more accurate? If so, say which one. If not, explain why.

b. Is it possible to tell which thermometer is more precise? If so, say which one. If not, explain why.

Assume that X and Y are independent measurements with uncertainties σ_{X} = 0.3 and σ_{Y} = 0.2. Find the uncertainties in the following quantities:

a. 4X

b. X + 2Y

c. 2X − 3Y

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 + Z

b. Z/(X + Y )

c. √X( ln Y + Z)

d. Xe^{Z}^{2−2}Y

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. Accurate but not precise

ii. Precise but not accurate

iii. Neither accurate nor precise

iv. Both accurate and precise

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 uncertainty in each of the following quantities:

a. XY Z

b. √XY^{2}Z^{3}

c. (XY/Z)

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?

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 Research, 2002:142–148) discusses experiments to determine the effect of the solids concentration on the performance of treatment methods for wastewater sludge. In the first experiment, the concentration of solids (in g/L) was 43.94 ± 1.18. In the second experiment, which was independent of the first, the concentration was 48.66 ± 1.76. Estimate the difference in the concentration between the two experiments, and find the uncertainty in the estimate.

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 either l or d can be remeasured with relative uncertainty 0.2%. Which should be remeasured to provide the greater improvement in the relative uncertainty of the resistance?

Data from 14

The resistance R (in ohms) of a cylindrical conductor is given by R = kl/d^{2}, 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 = 4.4 ± 0.1 cm.

Let X ∼ Bin(7, 0.3). Find

a. P(X = 1)

b. P(X = 2)

c. P(X < 1)

d. P(X > 4)

e. μ_{X}

f. σ^{2}_{X}

Two resistors, with resistances R_{1} and R_{2}, are connected in series. R_{1} is normally distributed with mean 100 Ω and standard deviation 5 Ω, and R_{2} is normally distributed with mean 120 Ω and standard deviation 10 Ω.

a. What is the probability that R_{2} > R_{1}?

b. What is the probability that R_{2} exceeds R_{1} by more than 30 Ω?

Refer to Example 4.27. Estimate the probability that a 1 m_{2} sheet of aluminum has exactly one flaw, and find the uncertainty in this estimate.

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 × 10^{23} molecules). If X is the molarity of a solution of sodium chloride (NaCl), and Y is the molarity of a solution of sodium carbonate (Na_{2}CO_{3}), the molarity of sodium ion (Na^{+}) in a solution made of equal parts NaCl and Na_{2}CO_{3} is given by M = 0.5X + Y. Assume X and Y are independent and normally distributed, and that X has mean 0.450 and standard deviation 0.050, and Y has mean 0.250 and standard deviation 0.025.

a. What is the distribution of M?

b. Find P(M > 0.5).

Let X ∼ Bin(n, p), and let Y = n − X. Show that Y ∼ Bin(n, 1 − p).

An alternative to the lognormal distribution for modeling highly skewed populations is the Pareto distribution with parameters Î¸ and r. The probability density function is

The parameters Î¸ and r may be any positive numbers. Let X be a random variable with this distribution.

a. Find the cumulative distribution function of X.

b. Assume r > 1. Find Î¼_{X}.

c. Assume r > 2. Find Ïƒ^{2}_{X}.

d. Show that if r â‰¤ 1, Î¼_{X} does not exist.

e. Show that if r â‰¤ 2, Ïƒ^{2}_{X} does not exist.

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 the receiver concludes that m = 0, and if X > 0.5, then the receiver concludes that m = 1.

a. If the true message is m = 0, what is the probability of an error, that is, what is the probability that the receiver concludes that m = 1?

b. If the true message is m = 1, what is the probability of an error, that is, what is the probability that the receiver concludes that m = 0?

c. A string consisting of 60 1s and 40 0s will be sent. A bit is chosen at random from this string. What is the probability that it will be received correctly?

d. Refer to part (c). A bit is chosen at random from the received string. Given that this bit is 1, what is the probability that the bit sent was 0?

e. Refer to part (c). A bit is chosen at random from the received string. Given that this bit is 0, what is the probability that the bit sent was 1?

Refer to Exercise 23.

A binary message m, where m is equal either to 0 or to 1, is sent over an information channel. Because of noise in the channel, the message received is X, where X = m + E, and E is a random variable representing the channel noise. Assume that if X ≤ 0.5 then the receiver concludes that m = 0 and that if X > 0.5 then the receiver concludes that m = 1. Assume that E ∼ N(0, 0.25).

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, and the system will work if at least two of the four components work. If the probability that a component works is 0.9, and if the components function independently, which design has the greater probability of functioning?

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 correctly, the adhesive strength X is normally distributed with a mean of 200 N and a standard deviation of 10 N. Each hour, you make one measurement of the adhesive strength. You are supposed to inform your supervisor if your measurement indicates that the process has strayed from its target distribution.

a. Find P(X ≤ 160), under the assumption that the process is functioning correctly.

b. Based on your answer to part (a), if the process is functioning correctly, would a strength of 160 N be unusually small? Explain.

c. If you observed an adhesive strength of 160 N, would this be convincing evidence that the process was no longer functioning correctly? Explain.

d. Find P(X ≥ 203), under the assumption that the process is functioning correctly.

e. Based on your answer to part (d), if the process is functioning correctly, would a strength of 203 N be unusually large? Explain.

f. If you observed an adhesive strength of 203 N, would this be convincing evidence that the process was no longer functioning correctly? Explain.

g. Find P(X ≤ 195), under the assumption that the process is functioning correctly.

h. Based on your answer to part (g), if the process is functioning correctly, would a strength of 195 N be unusually small? Explain.

i. If you observed an adhesive strength of 195 N, would this be convincing evidence that the process was no longer functioning correctly? Explain.

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 restaurants with health code violations will be visited?

b. What is the probability that none of the restaurants that are visited will have health code violations?

Radioactive mass 1 emits particles at a mean rate of Î»_{1}per second, and radioactive mass 2 emits particles at a mean rate of Î»_{2}per second. Mass 1 is selected with probability p, and mass 2 is selected with probability 1 âˆ’ p. Let X be the time at which the first particle is emitted. It can be shown that X has a mixed exponential distribution with probability density function

a. Find Î¼_{X}.

b. Find the cumulative distribution function of X.

c. Let Î»_{1} = 2, Î»_{2} = 1, and p = 0.5. Find P(X â‰¤ 2).

d. Let Î»_{1} = 2, Î»_{2} = 1, and p = 0.5. Given that.

P(X â‰¤ 2), find the probability that mass 1 was selected.

Chebyshev’s inequality states that for any random variable X with mean μ and variance σ^{2}, and for any positive number k, P(|X − μ| ≥ kσ) ≤ 1/k^{2}. Let X ∼ N(μ, σ^{2}). Compute P(|X −μ| ≥ kσ) for the values k = 1, 2, and 3. Are the actual probabilities close to the Chebyshev bound of 1/k^{2}, or are they much smaller?

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.

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 formed in this way.

Let X âˆ¼ Bin(n, p).

a. Show that if x is an integer between 1 and n inclusive, then

b. Show that if X âˆ¼ Bin(n, p), the most probable value for X is the greatest integer less than or equal to np + p. Use part (a) to show that P(X = x) â‰¥ P(X = x âˆ’ 1) if and only if x â‰¤ np + p.

Let X âˆ¼ Poisson(Î»).

a. Show that if x is a positive integer, thenb. Show that if X âˆ¼ Poisson(Î»), the most probable value for X is the greatest integer less than or equal to Î». Use part (a) to show that P(X = x) â‰¥ P(X = x âˆ’1) if and only if x â‰¤ Î».

a. Show that if x is a positive integer, thenb. Show that if X âˆ¼ Poisson(Î»), the most probable value for X is the greatest integer less than or equal to Î». Use part (a) to show that P(X = x) â‰¥ P(X = x âˆ’1) if and only if x â‰¤ Î».

Let Z âˆ¼ 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 Ï† represent the probability density function, so

a. Show that the cumulative distribution function of

b. Differentiate F_{X} (x) to show that X âˆ¼ N(Î¼, Ïƒ^{2}).

c. Now let X = âˆ’Ïƒ Z + Î¼. Compute the cumulative distribution function of X in terms of Î¦, then differentiate it to show that X âˆ¼ N(Î¼, Ïƒ^{2}).

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 proportion of customers whose service was interrupted one or more times in the past month.

b. Find a 99% confidence interval for the proportion of customers whose service was interrupted one or more times in the past month.

c. Find the sample size needed for a 95% confidence interval to specify the proportion to within ± 0.05.

d. Find the sample size needed for a 99% confidence interval to specify the proportion to within ± 0.05.

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 10◦C/s, cooling all 10 welds at the rate of 30◦C/s, or cooling 5 welds at 10◦C/s and 5 at 30◦C/s? Explain.

Refer to Exercise 11.

In a study of the effect of cooling rate on the hardness of welded joints, 50 welds cooled at a rate of 10◦C/s had an average Rockwell (B) hardness of 91.1 and a standard deviation of 6.23, and 40 welds cooled at a rate of 30◦C/s had an average hardness of 90.7 and a standard deviation of 4.34.

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 made?

Refer to Exercise 5.

In a sample of 100 steel wires the average breaking strength is 50 kN, with a standard deviation of 2 kN.

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 study of sleep habits in a large number of subjects. In a sample of 87 young adults, the average time per day spent in bed (either awake or asleep) was 7.70 hours, with a standard deviation of 1.02 hours, and the average time spent in bed asleep was 7.06 hours, with a standard deviation of 1.11 hours. The mean time spent in bed awake was estimated to be 7.70−7.06 = 0.64 hours. Is it possible to compute a 95% confidence interval for the mean time spent in bed awake? If so, construct the confidence interval. If not possible, explain why not.

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 interval for the mean sugar content.

b. Find a 99% confidence interval for the mean sugar content.

c. What is the confidence level of the interval (11.81, 11.99)?

d. How large a sample is needed so that a 95% confidence interval specifies the mean to within±0.1?

e. How large a sample is needed so that a 99% confidence interval specifies the mean to within±0.1?

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 of confidence can this statement be made?

Refer to Exercise 7.

The capacities (in ampere-hours) were measured for a sample of 120 batteries. The average was 178 and the standard deviation was 14.

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, and B. Nyquist, Journal of Transportation Engineering 2011:277–286) defines the “indecision zone” as the period when a vehicle is between 2.5 and 5.5 seconds away from an intersection. It presents observations of 710 vehicles passing through various intersections in Vermont for which the light turned yellow in the indecision zone. Of the 710 vehicles, 89 ran a red light.

a. Find a 90% confidence interval for the proportion of vehicles that will run the red light when encountering a yellow light in the indecision zone.

b. Find a 95% confidence interval for the proportion of vehicles that will run the red light when encountering a yellow light in the indecision zone.

c. Find a 99% confidence interval for the proportion of vehicles that will run the red light when encountering a yellow light in the indecision zone.

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 Anatolia” (D. Whitney, P. Umhoefer, et al., Turkish Journal of Earth Sciences, 2008:209–217) reports that fifteen tracks on one rock specimen had an average track length of 13 μm with a standard deviation of 2 μm. Assuming this to be a random sample from an approximately normal population, find a 99% confidence interval for the mean track length for this rock specimen.

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 interval for the mean efficiency.

c. What is the confidence level of the interval (84.63, 85.37)?

d. How many thermostats must be sampled so that a 95% confidence interval specifies the mean to within ±0.35?

e. How many thermostats must be sampled so that a 99.5% confidence interval specifies the mean to within ±0.35?

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 were found to have a mean lifetime of 1827 hours and a standard deviation of 168 hours. A sample of 180 transistors of type B were tested and were found to have a mean lifetime of 1658 hours and a standard deviation of 225 hours. Find a 95% confidence interval for the difference between the mean lifetimes of the two types of transistors.

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. Should the t curve be used to find a 99% confidence interval for the thickness? If so, find the confidence interval. If not, explain why not.

c. Six independent measurements are taken of the thickness of another piece of sheet metal. The measurements this time are: 1.317, 1.318, 1.301, 1.307, 1.374, 1.323. Make a dotplot of these values.

d. Should the t curve be used to find a 95% confidence interval for the thickness of this metal? If so, find the confidence interval. If not, explain why not.

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 deviation was 5.1◦F.

a. Find a 90% confidence interval for the mean oven temperature.

b. Find a 95% confidence interval for the mean oven temperature.

c. What is the confidence level of the interval (347.5, 348.9)?

d. How many thermostats must be sampled so that a 90% confidence interval specifies the mean to within ± 0.8◦F?

e. How many thermostats must be sampled so that a 95% confidence interval specifies the mean to within ± 0.8◦F?

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 statement be made?

Refer to Exercise 8.

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 deviation was 5.1◦F.

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 Quitting: Differences by Race/Ethnicity” (E. Lloyd-Richardson, C. Stanton, et al., Am J Health Behav, 2008:3–15) surveyed 444 HIV-positive smokers. Of these, 170 reported that they had used a nicotine patch. Consider this to be a simple random sample.

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 deviation of 350 psi, and a random sample of 90 joints from species B had a mean shear stress of 1400 psi and a standard deviation of 250 psi. Find a 98% confidence interval for the difference in mean shear stress between the two species.

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 Journal of Veterinary and Animal Science, 2005:1013–1018) reports that a sample of 87 one-year-old spotted flounder had an average length of 126.31 mm with a standard deviation of 18.10 mm, and a sample of 132 two-year-old spotted flounder had an average length of 162.41 mm with a standard deviation of 28.49 mm. Find a 95% confidence interval for the mean length increase between one-and two-year-old fish.

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 year.

a. Find a 95% confidence interval for the proportion of residences that reduced their water consumption.

b. Find a 99% confidence interval for the proportion of residences that reduced their water consumption.

c. Find the sample size needed for a 95% confidence interval to specify the proportion to within ± 0.05.

d. Find the sample size needed for a 99% confidence interval to specify the proportion to within ± 0.05.

e. Someone claims that more than 70% of residences reduced their water consumption. With what level of confidence can this statement be made?

f. If 95% confidence intervals are computed for 200 towns, what is the probability that more than 192 of the confidence intervals cover the true proportions?

Find the levels of the confidence intervals that have the following values of z_{α/2}:

a. z_{α/2} = 1.96

b. z_{α/2} = 2.17

c. z_{α/2} = 1.28

d. z_{α/2} = 3.28

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 traffic signals in heavily traveled intersections. The effectiveness of the new method was evaluated in a simulation study. In 50 simulations, the mean improvement in traffic flow in a particular intersection was 654.1 vehicles per hour, with a standard deviation of 311.7 vehicles per hour.

a. Find a 95% confidence interval for the improvement in traffic flow due to the new system.

b. Find a 98% confidence interval for the improvement in traffic flow due to the new system.

c. A traffic engineer states that the mean improvement is between 581.6 and 726.6 vehicles per hour.With what level of confidence can this statement be made?

d. Approximately what sample size is needed so that a 95% confidence interval will specify the mean to within ± 50 vehicles per hour?

e. Approximately what sample size is needed so that a 98% confidence interval will specify the mean to within ± 50 vehicles per hour?

Find the value of t_{n−1}_{,α} needed to construct an upper or lower confidence bound in each of the situations in Exercise 1.

Exercise 1.

Find the value of t_{n−1,α/2} needed to construct a two sided confidence interval of the given level with the given sample size:

a. Level 90%, sample size 12.

b. Level 95%, sample size 7.

c. Level 99%, sample size 2.

d. Level 95%, sample size 29.

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 which the weights of 296 eggs from 27 week-old breeding hens averaged 54.1 g with a standard deviation of 4.4 g, and weights of 296 eggs from 59 week-old hens averaged 72.7 g with a standard deviation of 4.7 g. Find a 95% confidence interval for the difference between the mean weights.

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 smokers. Of these, 170 reported that they had used a nicotine patch. Consider this to be a simple random sample.

a. Find a 95% confidence interval for the proportion of HIV-positive smokers who have used a nicotine patch.

b. Find a 99% confidence interval for the proportion of HIV-positive smokers who have used a nicotine patch.

c. Someone claims that the proportion is less than 0.40. With what level of confidence can this statement be made?

d. Find the sample size needed for a 95% confidence interval to specify the proportion to within ± 0.03.

e. Find the sample size needed for a 99% confidence interval to specify the proportion to within ± 0.03.

Refer to Exercise 1. Find a 95% lower confidence bound for the proportion of automobiles whose emissions exceed the standard.

Refer to Exercise 1

In a simple random sample of 70 automobiles registered in a certain state, 28 of them were found to have emission levels that exceed a state standard.

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.

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 Engineering Online, 2012) reports measurements of the total power, on the log scale, of the heart rate variability, in the frequency range 0.003 to 0.4 Hz, for a group of 40 patients aged 25–49 years and for a group of 43 patients aged 50–75 years. The mean for the patients aged 25–49 years was 3.64 with a standard deviation of 0.23, and the mean for the patients aged 50–75 years was 3.40 with a standard deviation of 0.28. Find a 95% confidence interval for the difference in mean power between the two age groups.

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 standard deviation of 9 pounds. A second group of 43 people went on the diet but didn’t exercise. After six months, their mean weight loss was 14 pounds, with a sample standard deviation of 7 pounds. Find a95%confidence interval for the mean difference between the weight losses.

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 surgeries of a certain

type, the average surgery time was 136.9 minutes with a standard deviation of 22.6 minutes.

a. Find a 95% confidence interval for the mean surgery time for this procedure.

b. Find a 99.5% confidence interval for the mean surgery time for this procedure.

c. A surgeon claims that the mean surgery time is between 133.9 and 139.9 minutes. With what level of confidence can this statement be made?

d. Approximately how many surgeries must be sampled so that a 95%confidence interval will specify the mean to within ± 3 minutes?

e. Approximately how many surgeries must be sampled so that a 99%confidence interval will specify the mean to within ± 3 minutes?

The following are summary statistics for a data set. Would it be appropriate to use the Studentâ€™s t distribution to construct a confidence interval from these data? Explain.

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 report, saying, “We are 95% confident that the mean cost of repairs is less than $160.” Is the supervisor underestimating the confidence, overestimating it, or getting it right? Explain.

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 light is 299,795 ± 1.96 km/s.

b. The probability is 95% that the speed of light is in the interval 299,795 ± 1.96.

c. If a 65th measurement is made, the probability is 95% that it would fall in the interval 299,795 ± 1.96.

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% confidence interval for the mean temperature at noon to be 57° ± (1.96)(20)/√365. Is this correct? Why or why not?

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 the best answer to fill in the blank: The interval from the sample of size 400 will be approximately as the interval from the sample of size 100.

i. One-eighth as wide

ii. One-fourth as wide

iii. One-half as wide

iv. The same width

v. Twice as wide

vi. Four times as wide

vii. Eight times as wide(iii) one-half as wide. The ratio of the widths is equal to the ratio of the standard deviations of the

sample mean, which is

Refer to Exercise 11.

a. Find a 95% upper confidence bound for the mean sugar content.

b. The claim is made that the mean sugar content is greater than 11.7 g.With what level of confidence can this statement be made?

Refer to Exercise 11.

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.

The concentration of carbon monoxide (CO) in a gas sample is measured by a spectrophotometer and found to be 85 ppm. Through long experience with this instrument, it is believed that its measurements are unbiased and normally distributed, with an uncertainty (standard deviation) of 8 ppm. Find a 95% confidence interval for the concentration of CO in this sample.

Refer to Exercise 9.

a. Find a 90% upper confidence bound for the mean weight.

b. Someone says that the mean weight is less than 1.585 g. With what level of confidence can this statement be made?

Refer to Exercise 9.

In a sample of 80 ten-penny nails, the average weight was 1.56 g and the standard deviation was 0.1 g.

A stock market analyst notices that in a certain year, the price of IBM stock increased on 131 out of 252 trading days. Can these data be used to find a 95% confidence interval for the proportion of days that IBM stock increases? Explain.

The article “A Music Key Detection Method Based on Pitch Class Distribution Theory” (J. Sun, H. Li, and L. Ma, International Journal of Knowledge-based and Intelligent Engineering Systems, 2011:165–175) describes a method of analyzing digital music files to determine the key in which the music is written. In a sample of 335 classical music selections, the key was identified correctly in 293 of them.

a. Find a 90% confidence interval for the proportion of pieces for which the key will be correctly identified.

b. How many music pieces should be sampled to specify the proportion to within ± 0.025 with 90% confidence?

c. Another method of key detection is to be tested. At this point, there is no estimate of the proportion of the time this method will be identified correctly. Find a conservative estimate of the sample size needed so that the proportion will be specified to within ± 0.03 with 90% confidence.

The following MINITAB output presents a confidence interval for a population mean.

a. How many degrees of freedom does the Studentâ€™s t distribution have?

b. Use the information in the output, along with the t table, to compute a 99% confidence interval.

In the article “Bactericidal Properties of Flat Surfaces and Nanoparticles Derivatized with Alkylated Polyethylenimines” (J. Lin, S. Qiu, et al., Biotechnology Progress, 2002:1082–1086), experiments were described in which alkylated polyethylenimines were attached to surfaces and to nanoparticles to make them bactericidal. In one series of experiments, the bactericidal efficiency against the bacterium E. coliwas compared for a methylated versus a nonmethylated polymer. The mean percentage of bacterial cells killed with the methylated polymer was 95 with a standard deviation of 1, and the mean percentage of bacterial cells killed with the nonmethylated polymer was 70 with a standard deviation of 6. Assume that five independent measurements were made on each type of polymer. Find a 95% confidence interval for the increase in bactericidal efficiency of the methylated polymer.

Stainless steels can be susceptible to stress corrosion cracking under certain conditions. A materials engineer is interested in determining the proportion of steel alloy failures that are due to stress corrosion cracking.

a. In the absence of preliminary data, how large a sample must be taken so as to be sure that a 98% confidence interval will specify the proportion to within ± 0.05?

b. In a sample of 200 failures, 30 of them were caused by stress corrosion cracking. Find a 98% confidence interval for the proportion of failures caused by stress corrosion cracking.

c. Based on the data in part (b), estimate the sample size needed so that the 98% confidence interval will specify the proportion to within ± 0.05.

Each day a quality engineer selects a random sample of 50 power supplies from the day’s production, measures their output voltages, and computes a 90% confidence interval for the mean output voltage of all the power supplies manufactured that day. What is the probability that more than 15 of the confidence intervals constructed in the next 200 days will fail to cover the true mean?

The article “Occurrence and Distribution of Ammonium in Iowa Groundwater” (K. Schilling, Water Environment Research, 2002:177–186) describes measurements of ammonium concentrations (in mg/L) at a large number of wells in the state of Iowa. These included 349 alluvial wells and 143 quaternary wells. The concentrations at the alluvial wells averaged 0.27 with a standard deviation of 0.40, and those at the quaternary wells averaged 1.62 with a standard deviation of 1.70. Find a 95% confidence interval for the difference in mean concentrations between alluvial and quaternary wells.

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