1 Million+ Step-by-step solutions

The article “Toward a Lifespan Metric of Reading Fluency” (S. Wallot and G. Van Orden, International Journal of Bifurcation and Chaos, 2011:1173–1192) described a study of reading speed for undergraduate and graduate students. In a sample of 24 undergraduates, the mean time to read a certain passage was 4.8 seconds, with a standard deviation of 1.9 seconds. In a sample of 24 Ph.D. students, the mean time was 2.8 seconds, with a standard deviation of 1.0 seconds. Find a 95% confidence interval for the difference in reading speed between the two groups.

A general method for finding a confidence interval for the difference between two means of normal populations is given by expression (5.21) on page 365. A pooled method that can be used when the variances of the populations are known to be equal is given by expression (5.22) on page 367. This exercise is designed to compare the coverage probabilities of these methods under a variety of conditions. A fair amount of coding may be required, depending on the software used.

a. Let n_{X} = 10, n_{Y} = 10, σ_{X} = 1, and σ_{Y} = 1. Generate 10,000 pairs of samples: X^{∗}_{1}, . . . , X^{∗}_{nX} from a N(0, σ^{2}_{X} ) distribution, and Y^{∗}_{1} , . . . , Y^{∗}_{nY} from a N(0, σ^{2}_{Y} ) distribution. For each pair of samples, compute a 95% confidence interval using the general method, and a 95% confidence interval using the pooled method. Note that each population has mean 0, so the true difference between the means is 0. Estimate the coverage probability for each method by computing the proportion of confidence intervals that cover the true value 0.

b. Repeat part (a), using n_{X} = 10, n_{Y} = 10, σ_{X} = 1, and σ_{Y} = 5.

c. Repeat part (a), using n_{X} = 5, n_{Y} = 20, σ_{X} = 1, and σ_{Y} = 5.

d. Repeat part (a), using n_{X} = 20, n_{Y} = 5, σ_{X} = 1, and σ_{Y} = 5.

e. Does the coverage probability for the general method differ substantially from 95% under any of the conditions in parts (a) through (d)? (It shouldn’t.)

f. Based on the results in parts (a) through (d), under which conditions does the pooled method perform most poorly?

i. When the sample sizes are equal and the variances differ.

ii. When both the sample sizes and the variances differ, and the larger sample comes from the population with the larger variance.

iii. When both the sample sizes and the variances differ, and the smaller sample comes from the population with the larger variance.

A pollster plans to survey a random sample of voters in a certain city to ask whether they support an increase in property taxes to fund the construction of a new elementary school. How many voters should be sampled to be sure that a 95% confidence interval for the proportion who favor the proposal specifies that proportion to within ± 0.04?

Polychlorinated biphenyls (PCBs) are a group of synthetic oil-like chemicals that were at one time widely used as insulation in electrical equipment and were discharged into rivers. They were discovered to be a health hazard and were banned in the 1970s. Since then, much effort has gone into monitoring PCB concentrations in waterways. Assume that water samples are being drawn from a waterway in order to estimate the PCB concentration.

a. Suppose that a random sample of size 80 has a sample mean of 1.69 ppb and a sample standard deviation of 0.25 ppb. Find a 95% confidence interval for the PCB concentration.

b. Estimate the sample size needed so that a 95% confidence interval will specify the population mean to within ± 0.02 ppb.

Ten samples of coal from a Northern Appalachian source had an average mercury content of 0.242 ppm with a standard deviation of 0.031 ppm. Find a 95% confidence for the mean mercury content of coal from this source.

A sample of 100 components is drawn, and a 95% confidence interval for the proportion defective specifies this proportion to within ± 0.06. To get a more precise estimate of the number defective, the sample size will be increased to 400, and the confidence interval will be recomputed. What will be the approximate width of the new confidence interval? Choose the best answer:

i. ± 0.015

ii. ± 0.03

iii. ± 0.06

iv. ± 0.12

v. ± 0.24

A computer sends a packet of information along a channel and waits for a return signal acknowledging that the packet has been received. If no acknowledgment is received within a certain time, the packet is re-sent. Let X represent the number of times the packet is sent. Assume that the probability mass function of X is given by

where c is a constant.

a. Find the value of the constant c so that p(x) is a probability mass function.

b. Find P(X = 2).

c. Find the mean number of times the packet is sent.

d. Find the variance of the number of times the packet is sent.

e. Find the standard deviation of the number of times the packet is sent.

For continuous random variables X and Y with joint probability density function.

a. Find P(X > 1 and Y > 1).

b. Find the marginal probability density functions f_{X} (x) and f_{Y} (y).

c. Are X and Y independent? Explain

Refer to Example 2.54.

a. Find Cov(X,Y).

b. Find ρ_{X,Y}.

The lifetimes, in months, of two components in a system, denoted X and Y , have joint probability density function.

a. What is the probability that both components last longer than one month?

b. Find the marginal probability density functions f_{X} (x) and f_{Y} (y).

c. Are X and Y independent? Explain.

A lot of 1000 components contains 300 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second component drawn is defective.

a. Find P(A).

b. Find P(B|A).

c. Find P(A ∩ B).

d. Find P(A^{c} ∩ B).

e. Find P(B).

f. Find P(A|B).

g. Are A and B independent? Is it reasonable to treat A and B as though they were independent? Explain.

If X is a random variable, prove that Cov(X, X) = σ^{2}_{X}.

The number of customers in line at a supermarket express checkout counter is a random variable whose probability mass function is given in the following table.

For each customer, the number of items to be purchased is a random variable with probability mass function.

Let X denote the number of customers in line, and let Y denote the total number of items purchased by all the customers in line. Assume the number of items purchased by one customer is independent of the number of items purchased by any other customer.

a. Find P(X = 2 and Y = 2).

b. Find P(X = 2 and Y = 6).

c. Find P(Y = 2).

Refer to Exercise 26.

a. Find Î¼_{X}.

b. Find Ïƒ^{2}_{X}.

c. Find Cov(X,Y).

d. Find Ï_{X,Y}.

a. Find Î¼

b. Find Ïƒ

c. Find Cov(X,Y).

d. Find Ï

Refer to Exercise 26.

The oxygen equivalence number of a weld is a number that can be used to predict properties such as hardness, strength, and ductility. The article “Advances in Oxygen Equivalence Equations for Predicting the Properties of Titanium Welds” (D. Harwig, W. Ittiwattana, and H. Castner, The Welding Journal, 2001:126s 136s) presents several equations for computing the oxygen equivalence number of a weld. An equation designed to predict the strength of a weld is X = 1.12C + 2.69N + O − 0.21 Fe, where X is the oxygen equivalence, and C, N, O, and Fe are the amounts of carbon, nitrogen, oxygen, and iron, respectively, in weight percent, in the weld. Suppose that for welds of a certain type, μ_{C} = 0.0247, μ_{N} = 0.0255, μ_{O} = 0.1668, μ_{Fe} = 0.0597, σ_{C} = 0.0131, σ_{N} = 0.0194, σ_{O} = 0.0340, and σ_{Fe} = 0.0413. Furthermore assume that correlations are given by ρ_{C,N} = −0.44, ρ_{C,O} = 0.58, ρ_{C,Fe} = 0.39, ρ_{N,O} = −0.32, ρ_{N,Fe} = 0.09, and ρ_{O,Fe} = −0.35.

a. Find μ_{X}.

b. Find Cov(C, N), Cov(C, O), Cov(C, Fe), Cov(N, O), Cov(N, Fe), and Cov(O, Fe).

c. Find σ_{X}.

Find the uncertainty in Y, given that X = 2.0 ± 0.3 and

a. Y = X^{3}

b. Y = √2X

c. Y = 3/X

d. Y = ln X

e. Y = e^{X}

f. Y = cos X (X is in units of radians)

The volume of a cone is given by V = πr^{2}h/3, where r is the radius of the base and h is the height. Assume the height is measured to be h = 6.00 ± 0.01 cm and the radius is r = 5.00 ± 0.02 cm.

a. Estimate the volume of the cone, and find the uncertainty in the estimate.

b. Which would provide a greater reduction in the uncertainty in V: reducing the uncertainty in h to 0.005 cm or reducing the uncertainty in r to 0.01 cm?

The volume of a cone is given by V = πr^{2}h/3, where r is the radius of the base and h is the height. Assume the radius is 5 cm, measured with negligible uncertainty, and the height is h = 6.00 ± 0.02 cm. Estimate the volume of the cone, and find the uncertainty in the estimate.

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.

Refer to Exercise 10 in Section 3.2. Assume that τ = 30.0 ± 0.1 Pa, h = 10.0 ± 0.2 mm, and μ = 1.49 Pa · s with negligible uncertainty.

a. Estimate V and find the uncertainty in the estimate.

b. Which would provide a greater reduction in the uncertainty in V: reducing the uncertainty in τ to 0.01 Pa or reducing the uncertainty in h to 0.1 mm?

Refer to Exercise 10

In the article “Temperature-Dependent Optical Constants of Water Ice in the Near Infrared: New Results and Critical Review of the Available Measurements” (B. Rajaram, D. Glandorf, et al., Applied Optics, 2001:4449–4462), the imaginary index of refraction of water ice is presented for various frequencies and temperatures. At a frequency of 372.1 cm−1 and a temperature of 166 K, the index is estimated to be 0.00116. At the same frequency and at a temperature of 196 K, the index is estimated to be 0.00129. The uncertainty is reported to be 10−4 for each of these two estimated indices. The ratio of the indices is estimated to be 0.00116/0.00129 = 0.899. Find the uncertainty in this ratio.

Convert the following absolute uncertainties to relative uncertainties.

a. 20.9 ± 0.4

b. 15.1 ± 0.8

c. 388 ± 23

d. 2.465 ± 0.009

In a chemical reaction run at a certain temperature, the concentration C of a certain reactant at time t is given by 1/C = kt+1/C_{0}, where C_{0} is the initial concentration and k is the rate constant. Assume the initial concentration is known to be 0.04 mol/L exactly. Assume that time is measured with negligible uncertainty.

a. After 30 s, the concentration C is measured to be 0.0038 ± 2.0 × 10^{−4} mol/L. Estimate the rate constant k, and find the relative uncertainty in the estimate.

b. After 50 s, the concentration C is measured to be 0.0024 ± 2.0×10^{−4} mol/L. Estimate the rate constant k and find the relative uncertainty in the estimate.

c. Denote the estimates of the rate constant k in parts (a) and (b) by k̂_{1} and k̂_{2}, respectively. The geometric mean √k_{1}k_{2} is used as an estimate of k. Find the relative uncertainty in this estimate

Refer to Exercise 16. Assume that T_{0}= 73.1 Â± 0.1Â°F, Ta = 37.5 Â± 0.2Â°F, k = 0.032 minâˆ’1 with negligible uncertainty, and T = 50Â°F exactly. Estimate t, and find the relative uncertainty in the estimate.

Refer to Exercise 16.

According to Newtonâ€™s law of cooling, the time t needed for an object at an initial temperature T_{0} to cool to a temperature T in an environment with ambient temperature T_{a }is given by

where k is a constant. Assume that for a certain type of container, k = 0.025 min^{âˆ’1}. Let t be the number of minutes needed to cool the container to a temperature of 50Â°F. Assume that T_{0} = 70.1 Â± 0.2Â°F and T_{a} = 35.7 Â± 0.1Â°F. Estimate t, and find the uncertainty in the estimate.

Given that X and Y are related by the given equation, and that X = 3.0 ± 0.1, estimate Y and its uncertainty.

a. XY = 1

b. Y/X = 2

c. √XY = 3

d. Y√X = 4

Refer to Exercise 5. Assume that the relative uncertainty in P_{1} is 5% and the relative uncertainty in P_{2} is 2%. Find the relative uncertainty in P_{3}.

Refer to Exercise 5.

When air enters a compressor at pressure P_{1} and leaves at pressure P_{2}, the intermediate pressure is given by P_{3} = √P_{1}P_{2}. Assume that P_{1} = 10.1 ± 0.3 MPa and P_{2} = 20.1 ± 0.4 MPa.

Of the items manufactured by a certain process, 20% are defective. Of the defective items, 60% can be repaired.

a. Find the probability that a randomly chosen item is defective and cannot be repaired.

b. Find the probability that exactly 2 of 20 randomly chosen items are defective and cannot be repaired.

Geologists estimate the time since the most recent cooling of a mineral by counting the number of uranium fission tracks on the surface of the mineral. A certain mineral specimen is of such an age that there should be an average of 6 tracks per cm^{2} of surface area. Assume the number of tracks in an area follows a Poisson distribution. Let X represent the number of tracks counted in 1 cm^{2} of surface area. Find

a. P(X = 7)

b. P(X ≥ 3)

c. P(2 < X < 7)

d. μ_{X}

e. σ_{X}

Below are the durations (in minutes) of 40 time intervals between eruptions of the geyser Old Faithful in Yellowstone National Park.

Construct a normal probability plot for these data. Do they appear to come from an approximately normal distribution?

Among all the income-tax forms filed in a certain year, the mean tax paid was $2000 and the standard deviation was $500. In addition, for 10% of the forms, the tax paid was greater than $3000. A random sample of 625 tax forms is drawn.

a. What is the probability that the average tax paid on the sample forms is greater than $1980?

b. What is the probability that more than 60 of the sampled forms have a tax of greater than $3000?

The lifetime of a laser (in hours) is lognormally distributed with μ = 8 and σ^{2} = 2.4. Two such lasers are operating independently.

a. Use a simulated sample of size 1000 to estimate the probability that the sum of the two lifetimes is greater than 20,000 hours.

b. Estimate the probability that both lasers last more than 3000 hours.

c. Estimate the probability that both lasers fail before 10,000 hours.

Refer to Exercise 4. What is the probability that in a sequence of 10 days, four green lights, one yellow light, and five red lights are encountered?

Refer to Exercise 4.

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 represents an independent trial.

In the article “Parameter Estimation with Only One Complete Failure Observation” (W. Pang, P. Leung, et al., International Journal of Reliability, Quality, and Safety Engineering, 2001:109–122), the lifetime, in hours, of a certain type of bearing is modeled with the Weibull distribution with parameters α = 2.25 and β = 4.474 × 10−4.

a. Find the probability that a bearing lasts more than 1000 hours.

b. Find the probability that a bearing lasts less than 2000 hours.

c. Find the median lifetime of a bearing.

d. The hazard function is defined in Exercise 8. What is the hazard at t = 2000 hours?

The temperature of a solution will be estimated by taking n independent readings and averaging them. Each reading is unbiased, with a standard deviation of σ = 0.5°C. How many readings must be taken so that the probability is 0.90 that the average is within ± 0.1°C of the actual temperature?

A random sample will be drawn from a normal distribution, for the purpose of estimating the population mean μ. Since μ is the median as well as the mean, it seems that both the sample median m and the sample mean X̅ are reasonable estimators. This exercise is designed to determine which of these estimators has the smaller uncertainty.

a. Compute the mean m̅^{∗} and the standard deviations m̅^{∗} of m^{∗}_{1}, . . ., m^{∗}_{1000}.

b. Compute the mean and standard deviation s_{X̅}^{∗} of X̅^{∗}_{1}, . . ., X̅^{∗}_{1000}.

c. The true value of μ is 0. Estimate the bias and uncertainty (σ_{m}) in m. In fact, the median is unbiased, so your bias estimate should be close to 0.

d. Estimate the bias and uncertainty (σ_{X̅}) in X̅. Is your bias estimate close to 0? Explain why it should be. Is your uncertainty estimate close to 1/√5? Explain why it should be.

Specifications for an aircraft bolt require that the ultimate tensile strength be at least 18 kN. It is known that 10% of the bolts have strengths less than 18.3 kN and that 5% of the bolts have strengths greater than 19.76 kN. It is also known that the strengths of these bolts are normally distributed.

a. Find the mean and standard deviation of the strengths.

b. What proportion of the bolts meet the strength specification?

Two-dimensional Poisson process. The number of plants of a certain species in a certain forest has a Poisson distribution with mean 10 plants per acre. The number of plants in T acres therefore has a Poisson distribution with mean 10T.

a. What is the probability that there will be exactly 18 plants in a two-acre region?

b. What is the probability that there will be exactly 12 plants in a circle with radius 100 ft? (1 acre = 43,560 ft^{2}.)

c. The number of plants of a different type follows a Poisson distribution with mean λ plants per acre, where λ is unknown. A total of 5 plants are counted in a 0.1 acre area. Estimate λ, and find the uncertainty in the estimate.

A process that polishes a mirrored surface leaves an average of 2 small flaws per 5 m^{2} of surface. The number of flaws on an area of surface follows a Poisson distribution.

a. What is the probability that a surface with area 3 m × 5 m will contain more than 5 flaws?

b. What is the probability that a surface with area 2 m × 3 m will contain no flaws?

c. What is the probability that 50 surfaces, each with dimensions 3 m × 6 m, will contain more than 350 flaws in total?

A machine produces 1000 steel O-rings per day. Each ring has probability 0.9 of meeting a thickness specification.

a. What is the probability that on a given day, fewer than 890 O-rings meet the specification?

b. Find the 60th percentile of the number of O-rings that meet the specification.

c. If the machine operates for five days, what is the probability that fewer than 890 O-rings meet the specification on three or more of those days?

A random sample of size 8 is taken from a Exp(λ) distribution, where λ is unknown. The sample values are 2.74, 6.41, 4.96, 1.65, 6.38, 0.19, 0.52, and 8.38. This exercise shows how to use the bootstrap to estimate the bias and uncertainty (σ_{λ̂}) in λ̂ = 1/X̅.

a. Compute λ̂ = 1/X̅ for the given sample.

b. Estimate the bias and uncertainty (λ̂ ^{*}_{1}) in λ̂ .

Of customers ordering a certain type of personal computer, 20% order an upgraded graphics card, 30% order extra memory, 15% order both the upgraded graphics card and extra memory, and 35% order neither. Fifteen orders are selected at random. Let X_{1}, X_{2}, X_{3}, X_{4} denote the respective numbers of orders in the four given categories.

a. Find P(X_{1} = 3, X_{2} = 4, X_{3} = 2, and X_{4 }= 6).

b. Find P(X_{1} = 3).

Let X ∼ U(a, b). Use the definition of the mean of a continuous random variable (Equation 2.35) to show that μ_{X} = (a + b)/2.

Shafts manufactured for use in optical storage devices have diameters that are normally distributed with mean μ = 0.652 cm and standard deviation σ = 0.003 cm. The specification for the shaft diameter is 0.650 ± 0.005 cm.

a. What proportion of the shafts manufactured by this process meet the specifications?

b. The process mean can be adjusted through calibration. If the mean is set to 0.650 cm, what proportion of the shafts will meet specifications?

c. If the mean is set to 0.650 cm, what must the standard deviation be so that 99% of the shafts will meet specifications?

The probability that a certain radioactive mass emits no particles in a one-minute time period is 0.1353. What is the mean number of particles emitted per minute?

Gears produced by a grinding process are categorized either as conforming (suitable for their intended purpose), downgraded (unsuitable for the intended purpose but usable for another purpose), or scrap (not usable). Suppose that 80% of the gears produced are conforming, 15% are downgraded, and 5% are scrap. Ten gears are selected at random.

a. What is the probability that one or more is scrap?

b. What is the probability that eight or more are not scrap?

c. What is the probability that more than two are either downgraded or scrap?

d. What is the probability that exactly nine are either conforming or downgraded?

A plate is attached to its base by 10 bolts. Each bolt is inspected before installation, and the probability of passing the inspection is 0.9. Only bolts that pass the inspection are installed. Let X denote the number of bolts that are inspected in order to attach one plate.

a. Find P(X = 12).

b. Find μ_{X}.

c. Find σ_{X}.

The concentration of particles in a suspension is 30 per mL.

a. What is the probability that a 2 mL sample will contain more than 50 particles?

b. Ten 2 mL samples are drawn. What is the probability that at least 9 of them contain more than 50 particles?

c. One hundred 2 mL samples are drawn. What is the probability that at least 90 of them contain more than 50 particles?

At a certain fast-food restaurant, 25% of drink orders are for a small drink, 35% for a medium, and 40% for a large. A random sample of 20 orders is selected for audit.

a. What is the probability that the numbers of orders for small, medium and large drinks are 5, 7, and 8, respectively?

b. What is the probability that more than 10 orders are for large drinks?

A thermocouple placed in a certain medium produces readings within 0.1°C of the true temperature 70% of the time, readings more than 0.1°C above the true temperature 10% of the time, and readings more than 0.1°C below the true temperature 20% of the time.

a. In a series of 10 independent readings, what is the probability that 5 are within 0.1°C of the true temperature, 2 are more than 0.1°C above, and 3 are more than 0.1°C below?

b. What is the probability that more than 8 of the readings are within 0.1°C of the true temperature?

The amount of paint required to paint a surface with an area of 50 m^{2} is normally distributed with mean 6 L and standard deviation 0.3 L.

a. If 6.2 L of paint are available, what is the probability that the entire surface can be painted?

b. How much paint is needed so that the probability is 0.9 that the entire surface can be painted?

c. What must the standard deviation be so that the probability is 0.9 that 6.2 L of paint will be sufficient to paint the entire surface?

Grandma is trying out a new recipe for raisin bread. Each batch of bread dough makes three loaves, and each loaf contains 20 slices of bread.

a. If she puts 100 raisins into a batch of dough, what is the probability that a randomly chosen slice of bread contains no raisins?

b. If she puts 200 raisins into a batch of dough, what is the probability that a randomly chosen slice of bread contains 5 raisins?

c. How many raisins must she put in so that the probability that a randomly chosen slice will have no raisins is 0.01?

A distributor receives a large shipment of components. The distributor would like to accept the shipment if 10% or fewer of the components are defective and to return it if more than 10% of the components are defective. She decides to sample 10 components, and to return the shipment if more than 1 of the 10 is defective.

a. If the proportion of defectives in the batch is in fact 10%, what is the probability that she will return the shipment?

b. If the proportion of defectives in the batch is 20%, what is the probability that she will return the shipment?

c. If the proportion of defectives in the batch is 2%, what is the probability that she will return the shipment?

d. The distributor decides that she will accept the shipment only if none of the sampled items are defective. What is the minimum number of items she should sample if she wants to have a probability no greater than 0.01 of accepting the shipment if 20% of the components in the shipment are defective?

The lifetime of a microprocessor is exponentially distributed with mean 3000 hours.

a. What proportion of microprocessors will fail within 300 hours?

b. What proportion of microprocessors will function for more than 6000 hours?

c. A new microprocessor is installed alongside one that has been functioning for 1000 hours. Assume the two microprocessors function independently. What is the probability that the new one fails before the old one?

A battery manufacturer claims that the lifetime of a certain type of battery has a population mean of 40 hours and a standard deviation of 5 hours. Let X represent the mean lifetime of the batteries in a simple random sample of size 100.

a. If the claim is true, what is P(X ≤ 36.7)?

b. Based on the answer to part (a), if the claim is true, is a sample mean lifetime of 36.7 hours unusually short?

c. If the sample mean lifetime of the 100 batteries were 36.7 hours, would you find the manufacturer’s claim to be plausible? Explain.

d. If the claim is true, what is P(X ≤ 39.8)?

e. Based on the answer to part (d), if the claim is true, is a sample mean lifetime of 39.8 hours unusually short?

f. If the sample mean lifetime of the 100 batteries were 39.8 hours, would you find the manufacturer’s claim to be plausible? Explain.

Let U ∼ U(0, 1). Let a and b be constants with a < b, and let X = (b − a)U + a.

a. Find the cumulative distribution function of U

b. Show that P(X ≤ x) = P(U ≤ (x −a)/(b−a)).

c. Use the result of part (b) to show that X ∼ U(a, b).

The manufacture of a certain part requires two different machine operations. The time on machine 1 has mean 0.5 hours and standard deviation 0.4 hours. The time on machine 2 has mean 0.6 hours and standard deviation 0.5 hours. The times needed on the machines are independent. Suppose that 100 parts are manufactured.

a. What is the probability that the total time used by machine 1 is greater than 55 hours?

b. What is the probability that the total time used by machine 2 is less than 55 hours?

c. What is the probability that the total time used by both machines together is greater than 115 hours?

d. What is the probability that the total time used by machine 1 is greater than the total time used by machine 2?

Use the result of Exercise 17 and Table A.1 to find P(X = 10) where X ∼ Geom(0.3).

The area covered by 1 L of a certain stain is normally distributed with mean 10m^{2} and standard deviation 0.2m^{2}.

a. What is the probability that 1 L of stain will be enough to cover 10.3m^{2}?

b. What is the probability that 2 L of stain will be enough to cover 19.9m^{2}?

You have received a radioactive mass that is claimed to have a mean decay rate of at least 1 particle per second. If the mean decay rate is less than 1 per second, you may return the product for a refund. Let X be the number of decay events counted in 10 seconds.

a. If the mean decay rate is exactly 1 per second (so that the claim is true, but just barely), what is P(X ≤ 1)?

b. Based on the answer to part (a), if the mean decay rate is 1 particle per second, would one event in 10 seconds be an unusually small number?

c. If you counted one decay event in 10 seconds, would this be convincing evidence that the product should be returned? Explain.

d. If the mean decay rate is exactly 1 per second, what is P(X ≤ 8)?

e. Based on the answer to part (d), if the mean decay rate is 1 particle per second, would eight events in 10 seconds be an unusually small number?

f. If you counted eight decay events in 10 seconds, would this be convincing evidence that the product should be returned? Explain.

Someone claims that the number of hits on his website has a Poisson distribution with mean 20 per hour. Let X be the number of hits in five hours.

a. If the claim is true, what is P(X ≤ 95)?

b. Based on the answer to part (a), if the claim is true, is 95 hits in a five-hour time period an unusually small number?

c. If you observed 95 hits in a five-hour time period, would this be convincing evidence that the claim is false? Explain.

d. If the claim is true, what is P(X ≤ 65)?

e. Based on the answer to part (d), if the claim is true, is 65 hits in a five-hour time period an unusually small number?

f. If you observed 65 hits in a five-hour time period, would this be convincing evidence that the claim is false? Explain.

Refer to Exercise 1. Another molecular biologist repeats the study with a different design. She makes up 12 DNA samples, and then chooses 6 at random to be treated with the enzyme and 6 to remain untreated. The results are as follows:

Find a 95% confidence interval for the difference between the mean numbers of fragments.

Refer to Exercise 1.

A molecular biologist is studying the effectiveness of a particular enzyme to digest a certain sequence of DNA nucleotides. He divides six DNA samples into two parts, treats one part with the enzyme, and leaves the other part untreated. He then uses a polymerase chain reaction assay to count the number of DNA fragments that contain the given sequence. The results are as follows:

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. 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. Since these standard deviations are based on large samples, assume they are negligibly different from the population standard deviations. An estimate for the ratio of the mean concentration in quaternary wells to the mean concentration in alluvial wells is R = 1.62/0.27 = 6.00.

a. Since the sample means 1.62 and 0.27 are based on large samples, it is reasonable to assume that they come from normal populations. Which distribution approximates the distribution of the sample mean concentration of alluvial wells, N(0.27, 0.402) or N(0.27, 0.402/349)? Which distribution approximates the distribution of the sample mean concentration of quaternary wells, N(1.62, 1.702) or N(1.62, 1.702/143)? Explain.

b. Generate a simulated sample of sample means and of ratios of sample means. Is it reasonable to assume that the ratio R is approximately normally distributed?

c. Use the simulated sample to estimate the standard deviation of R.

d. If appropriate, use the normal curve to find a 95% confidence interval for the ratio of the mean concentrations.

In a study of the lifetimes of electronic components, a random sample of 400 components are tested until they fail to function. The sample mean lifetime was 370 hours and the standard deviation was 650 hours. True or false:

a. An approximate 95% confidence interval for the mean lifetime of this type of component is from 306.3 to 433.7 hours.

b. About 95% of the sample components had lifetimes between 306.3 and 433.7 hours.

c. If someone takes a random sample of 400 components, divides the sample standard deviation of their lifetimes by 20, and then adds and subtracts that quantity from the sample mean, there is about a 68% chance that the interval so constructed will cover the mean lifetime of this type of component.

d. The z table can’t be used to construct confidence intervals here, because the lifetimes of the components don’t follow the normal curve.

e. About 68% of the components had lifetimes in the interval 370 ± 650 hours.

Boxes of nails contain 100 nails each. A sample of 10 boxes is drawn, and each of the boxes is weighed. The average weight is 1500 g and the standard deviation is 5 g. Assume the weight of the box itself is negligible, so that all the weight is due to the nails in the box.

a. Let μ_{box} denote the mean weight of a box of nails. Find a 95% confidence interval for μ_{box}.

b. Let μ_{nail} denote the mean weight of a nail. Express μ_{nai}_{l} in terms of μ_{box}.

c. Find a 95% confidence interval for μ_{nail}.

The answer to Exercise 19 part (d) is needed for this exercise. A geologist counts 64 emitted particles in one minute from a certain radioactive rock.

a. Find a 95% confidence interval for the rate of emissions in units of particles per minute.

b. After four minutes, 256 particles are counted. Find a 95% confidence interval for the rate of emissions in units of particles per minute.

c. For how many minutes should errors be counted in order that the 95% confidence interval specifies the rate to within ± 1 particle per minute?

The carbon content (in ppm) was measured for each of six silicon wafers. The results were

Assume that carbon contents are normally distributed.

a. Find a 95% prediction interval for the carbon content of a single wafer.

b. Find a tolerance interval for the carbon content that contains 90% of the wafers with 95% confidence.

Diameters, in mm, were measured for eight specimens of a certain type of ball bearing. The results were

Assume the diameters are normally distributed.

a. Find a 98% prediction interval for the diameter of a single ball bearing.

b. Find a tolerance interval for the diameter that contains 99% of the ball bearings with 95% confidence.

A sample of eight repair records for a certain fiber optic component was drawn, and the cost of each repair, in dollars, was recorded. The results were

Assume the population of repair records is normal.

Find a 95% confidence interval for the population standard deviation.

A test has power 0.80 when μ = 3.5. True or false:

a. The probability of rejecting H_{0} when μ = 3.5 is 0.80.

b. The probability of making a type I error when μ = 3.5 is 0.80.

c. The probability of making a type I error when μ = 3.5 is 0.20.

d. The probability of making a type II error when μ = 3.5 is 0.80.

e. The probability of making a type II error when μ = 3.5 is 0.20.

f. The probability that H_{0} is false when μ = 3.5 is 0.80.

A sample of 25 one-year-old girls had a mean weight of 24.1 pounds with a standard deviation of 4.3 pounds. Assume that the population of weights is normally distributed. A pediatrician claims that the standard deviation of the weights of one-year-old girls is less than 5 pounds. Do the data provide convincing evidence that the pediatrician’s claim is true? (Based on data from the National Health Statistics Reports.)

A copper smelting process is supposed to reduce the arsenic content of the copper to less than 1000 ppm. Let μ denote the mean arsenic content for copper treated by this process, and assume that the standard deviation of arsenic content is σ = 100 ppm. The sample mean arsenic content X̅ of 75 copper specimens will be computed, and the null hypothesis H_{0} : ≥ 1000 will be tested against the alternate H_{1} :μ < 1000.

a. A decision is made to reject H_{0} if X̅ ≤ 980. Find the level of this test.

b. Find the power of the test in part (a) if the true mean content is 965 ppm.

c. For what values of X̅ should H_{0} be rejected so that the power of the test will be 0.95 when the true mean content is 965?

d. For what values of X̅ should H_{0} be rejected so that the level of the test will be 5%?

e. What is the power of a 5% level test if the true mean content is 965 ppm?

f. How large a sample is needed so that a 5% level test has power 0.95 when the true mean content is 965 ppm?

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater than 85%. A random sample from a recent shipment of alumina yielded the following results (in percent):

A hypothesis test will be done to determine whether or not to accept the shipment.

a. State the appropriate null and alternate hypotheses.

b. Compute the P-value.

c. Should the shipment be accepted? Explain.

Refer to Exercise 1 in Section 5.2. Can it be concluded that less than half of the automobiles in the state have pollution levels that 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.

This exercise requires ideas from Section 2.6. In a two-sample experiment, when each item in one sample is paired with an item in the other, the paired t test (Section 6.8) can be used to test hypotheses regarding the difference between two population means. If one ignores the fact that the data are paired, one can use the two-sample t test (Section 6.7) as well. The question arises as to which test has the greater power. The following simulation experiment is designed to address this question.

Let (X_{1}, Y_{1}), . . . , (X_{8}, Y_{8}) be a random sample of eight pairs, with X_{1}, . . . , X_{8} drawn from an N(0, 1) population and Y_{1}, . . . , Y_{8} drawn from an N(1, 1) population. It is desired to test H_{0} :μ_{X} − μ_{Y} = 0 versus H_{1} :μ_{X} −μ_{Y} = 0. Note that μ_{X} = 0 and μ_{Y} = 1, so the true difference between the means is 1. Also note that the population variances are equal. If a test is to be made at the 5% significance level, which test has the greater power?

Let D_{i} = X_{i} − Y_{i} for i = 1, . . . , 10. The test statistic for the paired t test is D̅/(s_{D}/√8), where s_{D} is the standard deviation of the D_{i} (see Section 6.8). Its null distribution is Student’s t with seven degrees of freedom. Therefore the paired t test will reject H_{0} if | D̅/(s_{D}/√8)| > t_{7,.025 }= 2.365, so the power is √(|D̅/(s_{D}/√8)| > 2.365).

For the two-sample t test when the population variances are equal, the test statistic is D̅/(s_{p}√1/8 + 1/8) = D̅/(s_{p}/2), where s_{p} is the pooled standard deviation, which is equal in this case to p(s^{2}_{X}+ s^{2}_{Y} )/2. (See page 443. Note that D̅ =X –Y̅.) The null distribution is Student’s t with 14 degrees of freedom. Therefore the two-sample t test will reject H_{0} if | D̅/(s_{p}√1/8 + 1/8)| > t_{14,.025} = 2.145, and the power is P(|D̅/(s_{p}√1/8 + 1/8)| > 2.145). The power of these tests depends on the correlation between X_{i} and Y_{i} .

a. Generate 10,000 samples X^{∗}_{1i}, . . . , X^{∗}_{8i} froman N(0, 1) population and 10,000 samples Y^{∗}_{1i}, . . . , Y^{∗}_{8i} from an N(1, 1) population. The random variables X^{∗}_{ki} and Y^{∗}_{ki} are independent in this experiment, so their correlation is 0. For each sample, compute the test statistics D̅^{∗}/(s^{∗}_{D}/√8)and D̅^{∗}/(s^{∗}_{p}/2). Estimate the power of each test by computing the proportion of samples for which the test statistics exceeds its critical point (2.365 for the paired test, 2.145 for the two sample test). Which test has greater power?

b. As in part (a), generate 10,000 samples X^{∗}_{1i}, . . . , X^{∗}_{8i} from an N(0, 1) population. This time, instead of generating the values Y^{∗} independently, generate them so the correlation between X^{∗}_{ki} and Y^{∗}_{ki} is 0.8. This can be done as follows: Generate 10,000 samples Z^{∗}_{1i}, . . . , Z_{8i} from an N(0, 1) population, independent of the X^{∗} values. Then compute Y_{ki} = 1 + 0.8 X^{∗}_{ki} + 0.6 Z^{∗}_{ki}. The sample Y^{∗}_{1i}, . . . , Y^{∗}_{8i} will come from an N(1, 1) population, and the correlation between X^{∗}_{ki} and Y^{∗}_{ki} will be 0.8, which means that large values of X^{∗}_{ki} will tend to be paired with large values of Y^{∗}_{ki}, and vice versa. Compute the test statistics and estimate the power of both tests, as in part (a). Which test has greater power?

The article “Magma Interaction Processes Inferred from Fe-Ti Oxide Compositions in the D¨olek and Sari¸ci¸cek Plutons, Eastern Turkey” (O. Karsli, F. Aydin, et al., Turkish Journal of Earth Sciences, 2008:297–315) presents chemical compositions (in weight-percent) for several rock specimens. Fourteen specimens (two outliers were removed) of limenite grain had an average iron oxide (Fe_{2}O_{3}) content of 9.30 with a standard deviation of 2.71, and seven specimens of limenite lamella had an average iron oxide content of 9.47 with a standard deviation of 2.22. Can you conclude that the mean iron oxide content differs between limenite grain and limenite lamella?

At a certain genetic locus on a chromosome, each individual has one of three different DNA sequences (alleles). The three alleles are denoted A, B, C. At another genetic locus on the same chromosome, each organism has one of three alleles, denoted 1, 2, 3. Each individual therefore has one of nine possible allele pairs: A1, A2, A3, B1, B2, B3, C1, C2, or C3. These allele pairs are called haplotypes. The loci are said to be in linkage equilibrium if the two alleles in an individualâ€™s haplotype are independent. Haplotypes were determined for 316 individuals. The following MINITAB output presents the results of a chi-square test for independence.

Chi-Square Test: A, B, C Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts

a. How many individuals were observed to have the haplotype B3?

b. What is the expected number of individuals with the haplotype A2?

c. Which of the nine haplotypes was least frequently observed?

d. Which of the nine haplotypes has the smallest expected count?

e. Can you conclude that the loci are not in linkage equilibrium (i.e., not independent)? Explain.

f. Can you conclude that the loci are in linkage equilibrium (i.e., independent)? Explain.

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 H_{0} : μ = 1.4 versus H_{1} : μ = 1.4. The P-value will be____________.

i. Greater than 0.05

ii. Less than 0.05

iii. Equal to 0.05

Refer to Exercise 13 in Section 5.6. Can you conclude that the time to freeze-up is more variable in the seventh month than in the first month after installation?

Refer to Exercise 13

A windmill is used to generate direct current. Data are collected on 45 different days to determine the relationship between wind speed in mi/h (x) and current in kA (y). The data are presented in the following table.

a. Compute the least-squares line for predicting y from x. Make a plot of residuals versus fitted values.

b. Compute the least-squares line for predicting y from ln x. Make a plot of residuals versus fitted values.

c. Compute the least-squares line for predicting ln y from x. Make a plot of residuals versus fitted values.

The pressure of air (in MPa) entering a compressor is measured to be X = 8.5 ± 0.2, and the pressure of the air leaving the compressor is measured to be Y = 21.2 ± 0.3. The intermediate pressure is therefore measured to be P = √XY = 13.42. Assume that X and Y come from normal populations and are unbiased.

a. From what distributions is it appropriate to simulate values X^{∗} and Y^{∗}?

b. Construct a normal probability plot for the values P^{∗}. Is it reasonable to assume that P is approximately normally distributed?

The mass (in kg) of a soil specimen is measured to be X = 1.18 Â± 0.02. After the sample is dried in an oven, the mass of the dried soil is measured to be Y = 0.85 Â± 0.02. Assume that X and Y come from normal populations and are unbiased. The water content of the soil is measured to be

a. From what distributions is it appropriate to simulate values X^{âˆ—} and Y^{âˆ—}?

d. Construct a normal probability plot for the values W^{âˆ—}. Is it reasonable to assume that W is approximately normally distributed?

In Example 5.20 the following measurements were given for the cylindrical compressive strength (in MPa) for 11 beams:

One thousand bootstrap samples were generated from these data, and the bootstrap sample means were arranged in order. Refer to the smallest value as Y_{1}, the second smallest as Y_{2}, and so on, with the largest being Y_{1000}. Assume that Y_{25} = 38.3818, Y_{26} = 38.3818, Y_{50} = 38.3909, Y_{51} = 38.3918, Y_{950} = 38.5218, Y_{951} = 38.5236, Y_{975} = 38.5382, and Y_{976} = 38.5391.

a. Compute a 95% bootstrap confidence interval for the mean compressive strength, using method 1.

b. Compute a 95% bootstrap confidence interval for the mean compressive strength, using method 2.

c. Compute a 90% bootstrap confidence interval for the mean compressive strength, using method 1.

d. Compute a 90% bootstrap confidence interval for the mean compressive strength, using method 2.

Method 1:

Method 2:

A student measures the acceleration A of a cart moving down an inclined plane by measuring the time T that it takes the cart to travel 1 m and using the formula A = 2/T^{2}. Assume that T = 0.55 ± 0.01 s, and that the measurement T comes from a normal population and is unbiased.

Generate an appropriate simulated sample of values A^{∗}. Is it reasonable to assume that A is normally distributed?

Refer to Exercise 24. A sample of six repair records for a different type of component was drawn. The repair costs, in dollars, were as follows.

Would it be appropriate to compute a 95% confidence interval for the population standard deviation of the costs? If so, compute it. If not, explain why not.

The initial temperature of a certain container is measured to be T_{0}= 20Â°C. The ambient temperature is measured to be T_{a}= 4Â°C. An engineer uses Newtonâ€™s law of cooling to compute the time needed to cool the container to a temperature of 10Â°C. Taking into account the physical properties of the container, this time (in minutes) is computed to be

Assume that the temperature measurements T_{0} and T_{a }are unbiased and come from normal populations with standard deviation 0.1Â°C.

Generate an appropriate simulated sample of values T^{âˆ—}. Is it reasonable to assume that T is normally distributed?

A sample of seven concrete blocks had their crushing strength measured in MPa. The results were

Ten thousand bootstrap samples were generated from these data, and the bootstrap sample means were arranged in order. Refer to the smallest mean as Y_{1}, the second smallest as Y_{2}, and so on, with the largest being Y_{10,000}. Assume that Y_{50} = 1283.4, Y_{51} = 1283.4, Y_{100} = 1291.5, Y_{101} = 1291.5, Y_{250 }= 1305.5, Y_{251} = 1305.5, Y_{500} = 1318.5, Y_{501} = 1318.5, Y_{9500} = 1449.7, Y_{9501} = 1449.7, Y_{9750} = 1462.1, Y_{9751} = 1462.1, Y_{9900} = 1476.2, Y_{9901} = 1476.2, Y_{9950} = 1483.8, and Y_{9951} = 1483.8.

a. Compute a 95% bootstrap confidence interval for the mean compressive strength, using method 1.

b. Compute a 95% bootstrap confidence interval for the mean compressive strength, using method 2.

c. Compute a 99% bootstrap confidence interval for the mean compressive strength, using method 1.

d. Compute a 99% bootstrap confidence interval for the mean compressive strength, using method 2.

Refer to Exercise 4. Perform a randomization test to determine whether the mileage using regular gasoline has a greater variance than the mileage using premium gasoline. Generate at least 1000 random outcomes.

Two radon detectors were placed in different locations in the basement of a home. Each provided an hourly measurement of the radon concentration, in units of pCi/L. The data are presented in the following table.

a. Compute the least-squares line for predicting the radon concentration at location 2 from the concentration at location 1.

b. Plot the residuals versus the fitted values. Does the linear model seem appropriate?

c. Divide the data into two groups: points where R_{1} < 4 in one group, points where R_{1} â‰¥ 4 in the other. Compute the least-squares line and the residual plot for each group. Does the line describe either group well? Which one?

d. Explain why it might be a good idea to fit a linear model to part of these data, and a nonlinear model to the other.

During the production of boiler plate, test pieces are subjected to a load, and their elongations are measured. In one particular experiment, five tests will be made, at loads (in MPa) of 11, 37, 54, 70, and 93. The least-squares line will be computed to predict elongation from load. Confidence intervals for the mean elongation will be computed for several different loads. Which of the following intervals will be the widest? Which will be the narrowest?

i. The 95% confidence interval for the mean elongation under a load of 53 MPa.

ii. The 95% confidence interval for the mean elongation under a load of 72 MPa.

iii. The 95% confidence interval for the mean elongation under a load of 35 MPa.

A component can be manufactured according to either of two designs and with either a more expensive or a less expensive material. Several components are manufactured with each combination of design and material, and the lifetimes of each are measured (in hours). A two-way analysis of variance was performed to estimate the effects of design and material on component lifetime. The cell means and main effect estimates are presented in the following table.

The process engineer recommends that design 2 should be used along with the more expensive material. He argues that the main effects of both design 2 and the more expensive material are positive, so using this combination will result in the longest component life. Do you agree with the recommendation? Why or why not?

Refer to Exercise 6. The number of flaws in the 34th sample was 27. Is it possible to determine whether the process was in control at this time? If so, state whether or not the process was in control. If not, state what additional information would be required to make the determination.

In the following exercise, compute all partial derivatives.

v = 3x + 2xy^{4}

In the following exercise, compute all partial derivatives.

w = x^{3} + y^{3}/x^{2} + y^{2}

In the following exercise, compute all partial derivatives.

z = cos x sin y^{2}

In the following exercise, compute all partial derivatives.

v = e^{xy}

In the following exercise, compute all partial derivatives.

v = e^{x} (cos y + sin z)

In the following exercise, compute all partial derivatives.

w =√x^{2} + 4y^{2} + 3z^{2}

In the following exercise, compute all partial derivatives.

z = ln (x^{2} + y^{2})

In the following exercise, compute all partial derivatives.

v = e^{y2} cos(xz) + ln(x^{2} y + z)

In the following exercise, compute all partial derivatives.

v = 2xy^{3} − 3xy^{2}√xy

In the following exercise, compute all partial derivatives.

z = √sin(x^{2} y)

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