Get questions and answers for Essentials of Statistics

Imagine a researcher wanted to assess people’s fear of dogs as a function of the size of the dog. He assessed fear among people who indicated they were afraid of dogs, using a 30-point scale from 0 (no fear) to 30 (extreme fear). The researcher exposed each participant to three different dogs, a small dog weighing 20 pounds, a medium-sized dog weighing 55 pounds, and a large dog weighing 110 pounds, and assessed the fear level after each exposure. Here are some hypothetical data; note that these are the data from Exercise 11.39, on which you have already calculated numerous statistics:

a. State the null and research hypotheses.

b. Determine whether the assumptions of random selection and order effects were met.

c. In Exercise 11.39, you calculated the effect size for these data. What does this statistic tell us about the effect of size of dog on fear levels?

d. In Exercise 11.39, you calculated a Tukey HSD test for these data. What can you conclude about the effect of size of dog on fear levels based on this statistic? 

Commercials for chewing gum make claims about how long the flavor will last. In fact, some commercials claim that the flavor lasts too long, affecting sales and profit. Let’s put these claims to a test. Imagine a student decides to compare four different gums using five participants. Each randomly selected participant was asked to chew a different piece of gum each day for 4 days, such that at the end of the 4 days, each participant had chewed all four types of gum. The order of the gums was randomly determined for each participant. After 2 hours of chewing, participants recorded the intensity of flavor from 1 (not intense) to 9 (very intense). Here are some hypothetical data:

a. Conduct all six steps of the hypothesis test.

b. Are any additional tests warranted? Explain your answer.

Researchers Busseri, Choma, and Sadava (2009) asked a sample of individuals who scored as pessimists on a measure of life orientation about past, present, and projected future satisfaction with their lives. Higher scores on the life-satisfaction measure indicate higher satisfaction. The data below reproduce the pattern of means that the researchers observed in self-reported life satisfaction of the sample of pessimists for the three time points. Do pessimists predict a gloomy future for themselves?

a. Perform steps 5 and 6 of hypothesis testing. Be sure to complete the source table when calculating the F ratio for step 5.

b. If appropriate, calculate the Tukey HSD for all possible mean comparisons. Find the critical value of q and make a decision regarding the null hypothesis for each of the mean comparisons.

c. Calculate the R² measure of effect size for this ANOVA.

The previous exercise describes a study conducted by Busseri and colleagues (2009) using a group of pessimists. These researchers asked the same question of a group of optimists: Optimists rated their past, present, and projected future satisfaction with their lives. Higher scores on the life-satisfaction measure indicate higher satisfaction. The data below reproduce the pattern of means that the researchers observed in self-reported life satisfaction of the sample of optimists for the three time points. Do optimists see a rosy future ahead?

a. Perform steps 5 and 6 of hypothesis testing. Be sure to complete the source table when calculating the F ratio for step 5.

b. If appropriate, calculate the Tukey HSD for all possible mean comparisons. Find the critical value of q and make a decision regarding the null hypothesis for each of the mean comparisons.

c. Calculate the R² measure of effect size for this ANOVA. 

List the elements of each of the following sample spaces:

(a) The set of integers between 1 and 50 divisible by 8;

(b) The set S = {x | x² +4x−5=0 };

(c) The set of outcomes when a coin is tossed until a tail or three heads appear; 

(d) The set S = {x | x is a continent}; 

(e) The set S = {x | 2x−4 ≥ 0 and x<1}.

Two jurors are selected from 4 alternates to serve at a murder trial. Using the notation A₁A₃, for example, to denote the simple event that alternates 1 and 3 are selected, list the 6 elements of the sample space S.

Referring to Exercise 1.15 and the Venn diagram of Figure 1.7, list the numbers of the regions that represent the following events:

Fig 1.7

(a) The family will experience no mechanical problems and will not receive a ticket for a traffic violation but will arrive at a campsite with no vacancies. 

(b) The family will experience both mechanical problems and trouble in locating a campsite with a vacancy but will not receive a ticket for a traffic violation. 

(c) The family will either have mechanical trouble or arrive at a campsite with no vacancies but will not receive a ticket for a traffic violation. 

(d) The family will not arrive at a campsite with no vacancies.

Factory workers are constantly encouraged to practice zero tolerance when it comes to accidents in factories. Accidents can occur because the working environment or conditions themselves are unsafe. On the other hand, accidents can occur due to carelessness or so-called human error. In addition, the worker’s shift, 7:00 A.M.–3:00 P.M. (day shift), 3:00 P.M.–11:00 P.M. (evening shift), or 11:00 P.M.–7:00 A.M. (graveyard shift), may be a factor. During the last year, 300 accidents have occurred. The percentages of the accidents for the condition combinations are as follows:

If an accident report is selected randomly from the 300 reports, 

(a) What is the probability that the accident occurred on the graveyard shift? 

(b) What is the probability that the accident occurred due to human error? 

(c) What is the probability that the accident occurred due to unsafe conditions? 

(d) What is the probability that the accident occurred on either the evening or the graveyard shift?

Interest centers around the nature of an oven purchased at a particular department store. It can be either a gas or an electric oven. Consider the decisions made by six distinct customers.

(a) Suppose that the probability is 0.40 that at most two of these individuals purchase an electric oven. What is the probability that at least three purchase the electric oven?

(b) Suppose it is known that the probability that all six purchase the electric oven is 0.007 while 0.104 is the probability that all six purchase the gas oven. What is the probability that at least one of each type is purchased?

Prove that P(A'∩B') = 1 + P(A∩B) − P(A) − P(B).

If R is the event that a convict committed armed robbery and D is the event that the convict sold drugs, state in words what probabilities are expressed by

(a) P(R|D);

(b) P(D'|R);

(c) P(R'|D').

Pollution of the rivers in the United States has been a problem for many years. Consider the following events: A: the river is polluted, B: a sample of water tested detects pollution, C: fishing is permitted. Assume P(A) = 0.3, P(B|A) = 0.75, P(B|A') = 0.20, P(C|A∩B) = 0.20, P(C|A'∩B) = 0.15, P(C|A∩B') = 0.80, and P(C|A'∩B') = 0 .90. 

(a) Find P(A∩B∩C).  

(b) Find P(B'∩C). 

(c) Find P(C). 

(d) Find the probability that the river is polluted, given that fishing is permitted and the sample tested did not detect pollution.

Denote by A, B, and C the events that a grand prize is behind doors A, B, and C, respectively. Suppose you randomly picked a door, say A. The game host opened a door, say B, and showed there was no prize behind it. Now the host offers you the option of either staying at the door that you picked (A) or switching to the remaining unopened door (C). Use probability to explain whether you should switch or not.

An industrial plant is conducting a study to determine how quickly injured workers are back on the job following injury. Records show that 10% of all injured workers are admitted to the hospital for treatment and 15% are back on the job the next day. In addition, studies show that 2% are both admitted for hospital treatment and back on the job the next day. If a worker is injured, what is the probability that the worker will either be admitted to a hospital or be back on the job the next day or both?

An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function of T, the number of years to maturity for a randomly selected bond, is

find
(a) P(T = 5);
(b) P(T > 3);
(c) P(1.4 < T < 6);
(d) P(T ≤ 5 | T ≥ 2).

Consider the density function

(a) Evaluate k.
(b) Find F(x) and use it to evaluate P(0.3 < X < 0.6).

Suppose it is known from large amounts of historical data that X, the number of cars that arrive at a specific intersection during a 20-second time period, is characterized by the following discrete probability function:

(a) Find the probability that in a specific 20-second time period, more than 8 cars arrive at the intersection.
(b) Find the probability that only 2 cars arrive.

If the joint probability distribution of X and Y is given by

for x = 0, 1, 2, 3; y = 0, 1, 2,

find
(a) P(X ≤ 2, Y = 1);
(b) P(X >2, Y ≤ 1);
(c) P(X >Y);
(d) P(X + Y = 4).
(e) Find the marginal distribution of X; of Y .

From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. If X is the number of oranges and Y is the number of apples in the sample, find 

(a) The joint probability distribution of X and Y;
(b) P[(X, Y ) ∈ A], where A is the region that is given by {(x, y) | x + y ≤ 2};
(c) P(Y = 0 | X = 2);
(d) The conditional distribution of y, given X = 2.

A fast-food restaurant operates both a drive through facility and a walk-in facility. On a randomly selected day, let X and Y, respectively, be the proportions of the time that the drive-through and walk-in facilities are in use, and suppose that the joint density function of these random variables is

(a) Find the marginal density of X.
(b) Find the marginal density of Y .
(c) Find the probability that the drive-through facility is busy less than one-half of the time.

Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (^â—¦F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density

Find

(a) 

(b) P(X

Each rear tire on an experimental airplane is supposed to be filled to a pressure of 40 pounds per square inch (psi). Let X denote the actual air pressure for the right tire and Y denote the actual air pressure for the left tire. Suppose that X and Y are random variables with the joint density function

(a) Find k.
(b) Find P(30 ≤ X ≤ 40 and 40 ≤Y <50).
(c) Find the probability that both tires are underfilled.

Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y have the joint density

Find P(X +Y >1/2).

Let X denote the number of times a certain numerical control machine will malfunction: 1, 2, or 3 times on any given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability distribution is given as

(a) Evaluate the marginal distribution of X.
(b) Evaluate the marginal distribution of Y.
(c) Find P(Y = 3 | X = 2).

Given the joint density function

find P(1 < Y < 3 | X = 1).

The joint probability density function of the random variables X, Y , and Z is

Find
(a) the joint marginal density function of Y and Z;
(b) the marginal density of Y;
(c) 

(d)

The density function of the continuous random variable X, the total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year, is given in Exercise 2.7 

Find the average number of hours per year that families run their vacuum cleaners.

Suppose that you are inspecting a lot of 1000 light bulbs, among which 20 are defectives. You choose two light bulbs randomly from the lot without replacement.Let

Find the probability that at least one light bulb chosen is defective.

A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous year. Suppose that the number of word processors, X, purchased each year has the following probability distribution:

If the cost of the desired model is $1200 per unit and at the end of the year a refund of 50X² dollars will be issued, how much can this firm expect to spend on new word processors during this year?

For the random variables X and Y whose joint density function is given in Exercise 2.32 on page 72, find the covarianc

Exercise 2.32

A fast-food restaurant operates both a drive through facility and a walk-in facility. On a randomly selected day, let X and Y, respectively, be the proportions of the time that the drive-through and walk-in facilities are in use, and suppose that the joint density function of these random variables is

For the random variables X and Y in Exercise 2.31 on page 72, determine the correlation coefficient between X and Y.

Exercise 2.31

From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. If X is the number of oranges and Y is the number of apples in the sample.

Random variables X and Y follow a joint distribution

Determine the correlation coefficient between X and Y .

Repeat Exercise 2.83 on page 88 by applying Theorem 2.5 and Corollary 2.6.

Exercise 2.83

The length of time, in minutes, for an airplane to obtain clearance for takeoff at a certain airport is a random variable Y = 3X −2, where X has the density function

If a random variable X is defined such that E[(X − 1)²] = 10 and E[(X − 2)²] = 6, find μ and σ².

The total time, measured in units of 100 hours, that a teenager runs her hair dryer over a period of one year is a continuous random variable X that has the density function

Use Theorem 2.6 to evaluate the mean of the random variable Y = 60X² + 39X, where Y is equal to the number of kilowatt hours expended annually.

Another type of system that is employed in engineering work is a group of parallel components or a parallel system. In this more conservative approach, the probability that the system operates is larger than the probability that any component operates. The system fails only when all components fail. Consider a situation in which there are 4 independent components in a parallel system with probability of operation given by Component 1: 0.95; Component 2: 0.94; Component 3: 0.90; Component 4: 0.97. What is the probability that the system does not fail?

Consider Exercise 2.58 on page 79. Can it be said that the ratings given by the two experts are independent? Explain why or why not.

Exercise 2.58

Two tire-quality experts examine stacks of tires and assign a quality rating to each tire on a 3-point scale. Let X denote the rating given by expert A and Y denote the rating given by B. The following table gives the joint distribution for X and Y.

In a support system in the U.S. space program, a single crucial component works only 85% of the time. In order to enhance the reliability of the system, it is decided that 3 components will be installed in parallel such that the system fails only if they all fail. Assume the components act independently and that they are equivalent in the sense that all 3 of them have an 85% success rate. Consider the random variable X as the number of components out of 3 that fail.
(a) Write out a probability function for the random variable X.
(b) What is E(X) (i.e., the mean number of components out of 3 that fail)?
(c) What is Var(X)?
(d) What is the probability that the entire system is successful?
(e) What is the probability that the system fails?
(f) If the desire is to have the system be successful with probability 0.99, are three components sufficient? If not, how many are required?

The purpose of the study The Incorporation of a Chelating Agent into a Flame Retardant Finish of a Cotton Flannelette and the Evaluation of Selected Fabric Properties, conducted at Virginia Tech, was to evaluate the use of a chelating agent as part of the flame retardant finish of cotton flannelette by determining its effect upon flammability after the fabric is laundered under specific conditions. There were two treatments at two levels. Two baths were prepared, one with carboxymethyl cellulose (bath I) and one without (bath II). Half of the fabric was laundered 5 times and half was laundered 10 times. There were 12 pieces of fabric in each bath/number of launderings combination. After the washings, the lengths of fabric that burned and the burn times were measured. Burn times (in seconds) were recorded as follows:

(a) Perform an analysis of variance. Is there a significant interaction term?
(b) Are there main effect differences? Discuss.

Personnel in the Materials Engineering Department at Virginia Tech conducted an experiment to study the effects of environmental factors on the stability of a certain type of copper-nickel alloy. The basic response was the fatigue life of the material. The factors are level of stress and environment. The data are as follows:

(a) Do an analysis of variance to test for interaction between the factors. Use α = 0.05.
(b) Based on part (a), do an analysis on the two main effects and draw conclusions. Use a P-value approach in drawing conclusions.

The printout in Figure 8.4 on page 373 gives information on Tukey’s test, using PROC GLM in SAS, for the aggregate data in Example 8.1. Give conclusions regarding paired comparisons using Tukey’s test results.

The study Loss of Nitrogen Through Sweat by Preadolescent Boys Consuming Three Levels of Dietary Protein was conducted by the Department of Human Nutrition and Foods at Virginia Tech to determine perspiration nitrogen loss at various dietary protein levels. Twelve preadolescent boys ranging in age from 7 years, 8 months to 9 years, 8 months, all judged to be clinically healthy, were used in the experiment. Each boy was subjected to one of three controlled diets in which 29, 54, or 84 grams of protein were consumed per day. The following data represent the body perspiration nitrogen loss, in milligrams, during the last two days of
the experimental period:

(a) Perform an analysis of variance at the 0.05 level of significance to show that the mean perspiration nitrogen losses at the three protein levels are different.
(b) Use Tukey’s test to determine which protein levels are significantly different from each other in mean nitrogen loss.

For the model of Exercise 7.49 on page 340, test the hypothesis that β₁= 2 against the alternative that β₁≠ 2. Use a P-value in your conclusion.

A study was performed on a type of bearing to find the relationship of amount of wear y to x₁= oil viscosity and x₂= load. The following data were obtained.

(Data from Myers, Montgomery, and Anderson-Cook, 2009.)
(a) Estimate the unknown parameters of the multiple linear regression equation

μ_Y |_x1,x2 = β₀ + β_1x1 + β_2x2.

(b) Predict wear when oil viscosity is 20 and load is 1200.

An experiment was conducted to study the size of squid eaten by sharks and tuna. The regressor variables are characteristics of the beaks of the squid. The data are given as follows:

In the study, the regressor variables and response considered are

x₁ = rostral length, in inches,
x₂ = wing length, in inches,
x₃ = rostral to notch length, in inches,
x₄ = notch to wing length, in inches,
x₅ = width, in inches,
y   = weight, in pounds.

Estimate the multiple linear regression equation

μ_Y |_{x1,x2,x3,x4,x5}
= Î²₀ + β₁x₁ + β₂x₂ + β₃x₃ + β₄x₄ + β₅x₅.

The following data reflect information from 17 U.S. Navy hospitals at various sites around the world. The regressors are workload variables, that is, items that result in the need for personnel in a hospital. A brief description of the variables is as follows:

y = monthly labor-hours,
x₁ = average daily patient load,
x₂ = monthly X-ray exposures,
x₃ = monthly occupied bed-days,
x₄ = eligible population in the area/1000,
x₅ = average length of patient’s stay, in days.

The goal here is to produce an empirical equation that will estimate (or predict) personnel needs for naval hos pitals. Estimate the multiple linear regression equation

μ_Y |_{x1,x2,x3,x4,x5}
= Î²₀ + β₁x₁ + β₂x₂ + β₃x₃ + β₄x₄ + β₅x₅.

The following data are given:

(a) Fit the cubic model Î¼_Y|_x = Î²₀ + β₁x + β₂x² + β₃x³.
(b) Predict Y when x = 2.

The following data represent the chemistry grades for a random sample of 12 freshmen at a certain college along with their scores on an intelligence test administered while they were still seniors in high school. the number of class periods missed is also given.

(a) Fit a multiple linear regression equation of the form yÌ‚ = b₀ + b₁x₁ + b₂x₂.
(b) Estimate the chemistry grade for a student who has an intelligence test score of 60 and missed 4 classes.

An experiment was conducted to determine if the weight of an animal can be predicted after a given period of time on the basis of the initial weight of the animal and the amount of feed that was eaten. Use the data, measured in kilograms, to do the followings. 

(a) Fit a multiple regression equation of the form

μ_{Y |x1,x2} = β₀ + β_1x1 + β_2x2.

(b) Predict the final weight of an animal having an initial weight of 35 kilograms that is given 250 kilograms of feed.

Organophosphate (OP) compounds are used as pesticides. However, it is important to study their effect on species that are exposed to them. In the laboratory study Some Effects of Organophosphate Pesticides on Wildlife Species, by the Department of Fisheries and Wildlife at Virginia Tech, an experiment was conducted in which different dosages of a particular OP pesticide were administered to 5 groups of 5 mice (Peromyscus leucopus). The 25 mice were females of similar age and condition. One group received no chemical. The basic response y was a measure of activity in the brain. It was postulated that brain activity would decrease with an increase in OP dosage. Use the provided data to do the following.
(a) Find the least squares estimates of β0 and β1 using the model

Y_i = β₀ + β₁x_i + ϵ_i,    i = 1, 2, . . . , 25.

(b) Construct an analysis-of-variance table in which the lack of fit and pure error have been separated. Determine if the lack of fit is significant at the 0.05 level. Interpret the results.

Use an analysis-of-variance approach to test the hypothesis that β₁ = 0 against the alternative hypothesis β₁ ≠ 0 in Exercise 7.5 on page 304 at the 0.05 level of significance.

Data From Exercise 7.5

Temperature, x        Converted Sugar, y
1.0.....................................8.1
1.1.....................................7.8
1.2.....................................8.5
1.3.....................................9.8
1.4.....................................9.5
1.5.....................................8.9
1.6.....................................8.6
1.7...................................10.2
1.8.....................................9.3
1.9.....................................9.2
2.0...................................10.5

Suppose we have a linear equation through the origin (Exercise 7.28) μ_Y|_x= β_x.
(a) Estimate the regression line passing through the origin for the following data:

(b) Suppose it is not known whether the true regression should pass through the origin. Estimate the linear model μ_{Y |x} = β₀ + β_1x and test the hypothesis that β₀ = 0, at the 0.10 level of significance, against the alternative that β₀ ≠ 0.

Test the hypothesis that β₁ = 6 in Exercise 7.9 on page 305 against the alternative that β₁ < 6. Use a 0.025 level of significance.

Data From Exercise 7.9

Advertising Costs ($)        Sales ($)
        40.................................. 385
        20.................................. 400
        25.................................. 395
        20.................................. 365
        30.................................. 475
        50.................................. 440
        40.................................. 490
        20.................................. 420
        50.................................. 560
        40.................................. 525
        25.................................. 480
        50.................................. 510

Test the hypothesis that β₀ = 10 in Exercise 7.8 on page 305 against the alternative that β₀ < 10. Use a 0.05 level of significance.
Data From Exercise 7.8

Placement Test         Course Grade
    50.................................. 53
    35.................................. 41
    35.................................. 61
    40.................................. 56
    55.................................. 68
    65.................................. 36
    35.................................. 11
    60.................................. 70
    90.................................. 79
    35.................................. 59
    90.................................. 54
    80.................................. 91
    60.................................. 48
    60.................................. 71
    60.................................. 71
    40.................................. 47
    55.................................. 53
    50.................................. 68
    65.................................. 57
    50.................................. 79

With reference to Exercise 7.3 on page 304,
(a) Evaluate s²;
(b) Construct a 99% confidence interval for β₀;
(c) Construct a 99% confidence interval for β₁.
Data From Exercise 7.3

With reference to Exercise 7.6 on page 304,
(a) Evaluate s²;
(b) Construct a 99% confidence interval for β₀;
(c) Construct a 99% confidence interval for β₁.

Data From Exercise 7.6

Normal Stress, x        Shear Resistance, y
26.8.......................................26.5
25.4.......................................27.3
28.9.......................................24.2
23.6.......................................27.1
27.7.......................................23.6
23.9.......................................25.9
24.7.......................................26.3
28.1.......................................22.5
26.9.......................................21.7
27.4.......................................21.4
22.6.......................................25.8
25.6.......................................24.9

With reference to Exercise 7.5 on page 304,
(a) Evaluate s²;
(b) Construct a 95% confidence interval for β₀;
(c) Construct a 95% confidence interval for β₁.

Data From Exercise 7.5

Temperature, x        Converted Sugar, y
1.0.....................................8.1
                                    1.1.....................................7.8                                    
1.2.....................................8.5
1.3.....................................9.8
1.4.....................................9.5
1.5.....................................8.9
1.6.....................................8.6
1.7.....................................10.2
1.8.....................................9.3
1.9.....................................9.2
2.0.....................................10.5

Physical fitness testing is an important aspect of athletic training. A common measure of the magnitude of cardiovascular fitness is the maximum volume of oxygen uptake during strenuous exercise. A study was conducted on 24 middle-aged men to determine the influence on oxygen uptake of the time required to complete a two-mile run. Oxygen uptake was measured with standard laboratory methods as the subjects performed on a treadmill. The work was published in “Maximal Oxygen Intake Prediction in Young and Middle Aged Males,” Journal of Sports Medicine  1969, 17–22. The data are as follows:

(a) Estimate the parameters in a simple linear regression model.
(b) Does the time it takes to run two miles have a significant influence on maximum oxygen uptake? Use H₀: β₁ = 0 versus H₁: β₁ ≠ 0.
(c) Plot the residuals on a graph against x and comment on the appropriateness of the simple linear model.

The Statistics Consulting Center at Virginia Tech analyzed data on normal woodchucks for the Department of Veterinary Medicine. The variables of interest were body weight in grams and heart weight in grams. It was desired to develop a linear regression equation in order to determine if there is a significant linear relationship between heart weight and total body weight. Use heart weight as the independent variable and body weight as the dependent variable and fit a simple linear regression using the following data. In addition, test the hypothesis H₀ : β₁ = 0 versus H₁ : β₁ = 0. Draw conclusions.

Body Weight (grams)            Heart Weight (grams)
4050.................................................11.2
2465.................................................12.4
3120.................................................10.5
5700.................................................13.2
2595.................................................9.8
3640.................................................11.0
2050.................................................10.8
4235.................................................10.4
2935.................................................12.2
4975.................................................11.2
3690.................................................10.8
2800.................................................14.2
2775.................................................12.2
2170.................................................10.0
2370.................................................12.3
2055.................................................12.5
2025.................................................11.8
2645.................................................16.0
2675.................................................13.8

Consider a 2nd-order response surface model that contains the linear, pure quadratic and cross product terms as follows:

y_i = β₀ + β₁x_1i + β₂x_2i + β₁₁x² _1i + Î²₂₂x² _2i + β₁₂x_1ix_2i + ϵ_i.

Fit the model above to the following data, and suggest any model editing that may be needed.

A study was conducted at Virginia Tech to determine if certain static arm-strength measures have an influence on the “dynamic lift” characteristics of an individual. Twenty-five individuals were subjected to strength tests and then were asked to perform a weightlifting test in which weight was dynamically lifted overhead. The data are given here.
(a) Estimate β₀and β₁for the linear regression curve μ_{Y |x}= β₀+ β_1x.
(b) Find a point estimate of Î¼_{Y |30}.
(c) Plot the residuals versus the x’s (arm strength).

Consider combinations of three factors in the removal of dirt from standard loads of laundry. The first factor is the brand of the detergent, X, Y , or Z. The second factor is the type of detergent, liquid or powder. The third factor is the temperature of the water, hot or warm. The experiment was replicated three times. Response is percent dirt removal. The data are as follows:

(a) Are there significant interaction effects at the α = 0.05 level?
(b) Are there significant differences between the three brands of detergent?
(c) Which combination of factors would you prefer to use?

State the null and alternative hypotheses to be used in testing the following claims and determine generally where the critical region is located:

(a) The mean snowfall at Lake George during the month of February is 21.8 centimeters.

(b) No more than 20% of the faculty at the local university contributed to the annual giving fund. 

(c) On the average, children attend schools within 6.2 kilometers of their homes in suburban St. Louis. 

(d) At least 70% of next year’s new cars will be in the compact and subcompact category.

(e) The proportion of voters favoring the incumbent in the upcoming election is 0.58.

(f) The average rib-eye steak at the Longhorn Steak house weighs at least 340 grams.

Construct a 95% confidence interval for σ² in Exercise 5.9 on page 213.

Exercise 5.9

Regular consumption of presweetened cereals contributes to tooth decay, heart disease, and other degenerative diseases, according to studies conducted by Dr. W. H. Bowen of the National Institute of Health and Dr. J. Yudben, Professor of Nutrition and Dietetics at the University of London. In a random sample consisting of 20 similar single servings of Alpha-Bits, the average sugar content was 11.3 grams with a standard deviation of 2.45 grams. Assuming that the sugar contents are normally distributed, construct a 95% confidence interval for the mean sugar content for single servings of Alpha-Bits.

A manufacturer of MP3 players conducts a set of comprehensive tests on the electrical functions of its product. All MP3 players must pass all tests prior to being sold. Of a random sample of 500 MP3 players, 15 failed one or more tests. Find a 90% confidence interval for the proportion of MP3 players from the population that pass all tests.

(a) A random sample of 200 voters in a town is selected, and 114 are found to support an annexation suit. Find the 96% confidence interval for the proportion of the voting population favoring the suit.
(b) What can we assert with 96% confidence about the possible size of our error if we estimate the proportion of voters favoring the annexation suit to be 0.57?

Compute 95% confidence intervals for the proportion of defective items in a process when it is found that a sample of size 100 yields 8 defectives.

Fortune magazine (March 1997) reported the total returns to investors for the 10 years prior to 1996 and also for 1996 for 431 companies. The total returns for 9 of the companies and the S&P 500 are listed below. Find a 95% confidence interval for the mean change in percent return to investors.

The federal government awarded grants to the agricultural departments of 9 universities to test the yield capabilities of two new varieties of wheat. Each variety was planted on a plot of equal area at each university, and the yields, in kilograms per plot, were recorded as follows:

Find a 95% confidence interval for the mean difference between the yields of the two varieties, assuming the differences of yields to be approximately normally distributed. Explain why pairing is necessary in this problem.

A taxi company is trying to decide whether to purchase brand A or brand B tires for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using 12 of each brand. The tires are run until they wear out. The results are

Brand A:    x̅₁ = 36,300 kilometers,
                   s₁ = 5000 kilometers.
Brand B:    x̅₂ = 38,100 kilometers,
                   s₂ = 6100 kilometers.

Compute a 95% confidence interval for μ_A − μ_B assuming the populations to be approximately normally distributed. You may not assume that the variances are equal.

Consider the situation of Case Study 5.1 with a larger sample of metal pieces. The diameters are as follows: 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 1.01, 1.03, 0.99, 1.00, 1.00, 0.99, 0.98, 1.01, 1.02, 0.99 centimeters. Once again the normality assumption may be made. Do the following and compare your results to those of the case study. Discuss how they are different and why.
(a) Compute a 99% confidence interval on the mean diameter.
(b) Compute a 99% prediction interval on the next diameter to be measured.
(c) Compute a 99% tolerance interval for coverage of the central 95% of the distribution of diameters.

A certain supplier manufactures a type of rubber mat that is sold to automotive companies. The material used to produce the mats must have certain hardness characteristics. Defective mats are occasionally discovered and rejected. The supplier claims that the proportion defective is 0.05. A challenge was made by one of the clients who purchased the mats, so an experiment was conducted in which 400 mats were tested and 17 were found defective.
(a) Compute a 95% two-sided confidence interval on the proportion defective.
(b) Compute an appropriate 95% one-sided confidence interval on the proportion defective.
(c) Interpret both intervals from (a) and (b) and comment on the claim made by the supplier.

A random sample of 25 tablets of buffered aspirin contains, on average, 325.05 mg of aspirin per tablet, with a standard deviation of 0.5 mg. Find the 95% tolerance limits that will contain 90% of the tablet contents for this brand of buffered aspirin. Assume that the aspirin content is normally distributed.

A manufacturer turns out a product item that is labeled either “defective” or “not defective.” In order to estimate the proportion defective, a random sample of 100 items is taken from production, and 10 are found to be defective. Following implementation of a quality improvement program, the experiment is conducted again. A new sample of 100 is taken, and this time only 6 are found to be defective. 

(a) Give a 95% confidence interval on p₁ − p₂, where p₁ is the population proportion defective before improvement and p₂ is the proportion defective after improvement. 

(b) Is there information in the confidence interval found in (a) that would suggest that p₁ > p₂? Explain.

Referring to Exercise 5.5, construct a 99% prediction interval for the kilometers traveled annually by an automobile owner in Virginia.

Exercise 5.5

A random sample of 100 automobile owners in the state of Virginia shows that an automobile is driven on average 23,500 kilometers per year with a standard deviation of 3900 kilometers. Assume the distribution of measurements to be approximately normal.

The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint:

Assuming that the measurements represent a random sample from a normal population, find a 95% prediction interval for the drying time for the next trial of the paint.

A labor union is becoming defensive about gross absenteeism by its members. The union leaders had always claimed that, in a typical month, 95% of its members were absent less than 10 hours. The union decided to check this by monitoring a random sample of 300 of its members. The number of hours absent was recorded for each of the 300 members. The results were x̅ = 6.5 hours and s = 2.5 hours. Use the data to respond to this claim, using a one-sided tolerance limit and choosing the confidence level to be 99%. Be sure to interpret what you learn from the tolerance limit calculation.

For a random sample X₁, . . . , X_n, show that

According to the Roanoke Times, in a particular year McDonald’s sold 42.1% of the market share of hamburgers. A random sample of 75 burgers sold resulted in 28 of them being from McDonald’s. Use material in Section 5.10 to determine if this information supports the claim in the Roanoke Times.

The concentration of an active ingredient in the output of a chemical reaction is strongly influenced by the catalyst that is used in the reaction. It is believed that when catalyst A is used, the population mean concentration exceeds 65%. The standard deviation is known to be σ= 5%. A sample of outputs from 30 independent experiments gives the average concentration of 

(a) Does this sample information with an average concentration of 

provide disturbing in-formation that perhaps μA is not 65%, but less than 65%? Support your answer with a probability statement.

(b) Suppose a similar experiment is done with the use of another catalyst, catalyst B. The standard deviation σ is still assumed to be 5% and X̅B turns outto be 70%. Comment on whether or not the sample information on catalyst B strongly suggests that μ_B is truly greater than μ_A. Support your answer by computing
(c) Under the condition thatμ_A = μ_B = 65%, give the approximate distribution of the following quantities (with mean and variance of each). Make use of the Central Limit Theorem.

i)

ii)

iii)

A taxi company tests a random sample of 10 steel-belted radial tires of a certain brand and records the following tread wear: 48,000, 53,000, 45,000, 61,000, 59,000, 56,000, 63,000, 49,000, 53,000, and 54,000 kilometers. The marketing claim for the tires is that, on the average, the tires last for 53,000 kilometers of use. In your answer, compute

and determine from Table A.4 (with 9 degrees of freedom) whether the computed t-value is reasonable or appears to be a rare event.

Consider Example 4.12 on page 189. Comment on any outliers.

Example 4.12

The following data represent the length of life in years, measured to the nearest tenth, of 30 similar fuel pumps:

Construct a box-and-whisker plot. Comment on the outliers in the data.

The following data represent the length of life,in seconds, of 50 fruit flies subject to a new spray in a controlled laboratory experiment:
Construct a box-and-whisker plot and comment on the nature of the sample. Compute the sample mean and sample standard deviation.

Consider Case Study 4.2 on page 170. Suppose 18 specimens were used for each type of paint in an experiment andthe actual difference in mean drying time, turned out to be 1.0.
(a) Does this seem to be a reasonable result if the two population mean drying times truly are equal? Make use of the result in the solution to Case Study 4.2.
(b) If someone did the experiment 10,000 times under the condition that μ_A=μ_B, in how many of those 10,000 experiments would there be a differencethat was as large as (or larger than)1.0?

For the data of Exercise 4.4, calculate the variance using the formula
(a) Of form (4.2.1);
(b) In Theorem 4.1.

Exercise 4.4

According to ecology writer Jacqueline Killeen, phosphates contained in household detergents pass right through our sewer systems, causing lakes to turn into swamps that eventually dry up into deserts. The following data show the amount of phosphates per load of laundry, in grams, for a random sample of various types of detergents used according to the prescribed directions:

National security requires that defense technology be able to detect incoming projectiles or missiles. To make the defense system successful, multiple radar screens are required. Suppose that three independent screens are to be operated and the probability that any one screen will detect an incoming missile is 0.8. Obviously, if no screens detect an incoming projectile, the system is untrustworthy and must be improved.
(a) What is the probability that an incoming missile will not be detected by any of the three screens?

(b) What is the probability that the missile will be detected by only one screen?
(c) What is the probability that it will be detected by at least two out of three screens?

Given a standard normal distribution, find the area under the curve that lies
(a) To the left of z = −1.39;
(b) To the right of z = 1.96;
(c) Between z = −2.16 and z = −0.65;
(d) To the left of z = 1.43;
(e) To the right of z = −0.89;
(f) Between z = −0.48 and z = 1.74.

Suppose X follows a continuous uniform distribution from 1 to 5. Determine the conditional probability P(X > 2.5 | X ≤ 4).

Given a continuous uniform distribution, show that
(a) μ = A + B/2 and
(b) σ² = (B − A)²/12 .

A company purchases large lots of a certain kind of electronic device. A method is used that rejects a lot if 2 or more defective units are found in a random sample of 100 units.
(a) What is the mean number of defective units found in a sample of 100 units if the lot is 1% defective?
(b) What is the variance?

Find the mean and variance of the random variable X in Exercise 3.40, representing the number of hurricanes per year to hit a certain area of the eastern United States.

Exercise 3.40

A certain area of the eastern United States is, on average, hit by 6 hurricanes a year. Find the prob-ability that in a given year that area will be hit by

On average, a textbook author makes two wordprocessing errors per page on the first draft of her textbook. What is the probability that on the next page she will make
(a) 4 or more errors?
(b) No errors?

What is the probability that a waitress will refuse to serve alcoholic beverages to only 2 minors if she randomly checks the IDs of 5 among 9 students, 4 of whom are minors?

According to USA Today (March 18, 1997), of 4 million workers in the general workforce, 5.8% tested positive for drugs. Of those testing positive, 22.5% were cocaine users and 54.4% marijuana users.
(a) What is the probability that of 10 workers testing positive, 2 are cocaine users, 5 are marijuana users, and 3 are users of other drugs?
(b) What is the probability that of 10 workers testing positive, all are marijuana users?
(c) What is the probability that of 10 workers testing positive, none is a cocaine user?

(a) In Exercise 3.7, how many of the 15 trucks would you expect to have blowouts?
(b) What is the variance of the number of blowouts experienced by the 15 trucks? What does that mean?

Exercise 3.7

In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the trucks fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability that

A random variable X that assumes the values x₁, x₂, . . . , x_k is called a discrete uniform random variable if its probability mass function is f(x) = 1/k for all of x₁, x₂, . . . , x_k and 0 otherwise. Find the mean and variance of X.

In a study conducted at Virginia Tech on the development of ectomycorrhizal, a symbiotic relationship between the roots of trees and a fungus, in which minerals are transferred from the fungus to the trees and sugars from the trees to the fungus, 20 northern red oak seedlings exposed to the fungus Pisolithus tinctorus were grown in a greenhouse. All seedlings were planted in the same type of soil and received the same amount of sunshine and water. Half received no ni222 trogen at planting time, to serve as a control, and the other half received 368 ppm of nitrogen in the form
NaNO3. The stem weights, in grams, at the end of 140 days were recorded as follows:

Construct a 95% confidence interval for the difference in the mean stem weight between seedlings that receive no nitrogen and those that receive 368 ppm of nitrogen. Assume the populations to be normally distributed with equal variances.

In Relief from Arthritis published by Thorsons Publishers, Ltd., John E. Croft claims that over 40% of those who suffer from osteoarthritis receive measurable relief from an ingredient produced by a particular species of mussel found off the coast of New Zealand. To test this claim, the mussel extract is to be given to a group of 7 osteoarthritic patients. If 3 or more of the patients receive relief, we shall not reject the null hypothesis that p = 0.4; otherwise, we shall conclude that p < 0.4.

(a) Evaluate α, assuming that p = 0.4.

(b) Evaluate β for the alternative p = 0.3.

The following scores represent the final examination grades for an elementary statistics course:

Test the goodness of fit between the observed class frequencies and the corresponding expected frequencies of a normal distribution with μ = 65 and σ = 21, using a 0.05 level of significance.