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Questions and Answers of
Statistics
A certain drug is suspected of lowering blood pressure as a side effect. A clinical trial is conducted to investigate this suspicion. Thirty-two patients are randomized into drug and placebo groups
The data for this question consist of a sample of 50 persons from the 1967-1980 Evans County Study (Schoenbach et al. 1986). Two basic independent variables are of interest: AGE and chronic disease
A researcher was interested in determining risk factors for high blood pressure (hypertension) among women. Data from a sample group of 680 women were collected. The following table gives the
A five-year follow-up study on 600 disease-free subjects was carried out to assess the effect of a (0-1) exposure variable E on the development or not of a certain disease. The variables AGE
A study was conducted on a sample of 53 patients presenting with prostate cancer who had also undergone a laparotomy to ascertain the extent of nodal involvement (Collett 1991). The result of the
A cross-sectional study was carried out to assess the relationship of alcohol and smoking to blood pressure in 2,500 men ages 20 years or older in four North American population groups, each group
A 2005 study was conducted to evaluate the influence of fear avoidance beliefs (FAB), chronicity of low back pain (CHR), and severity of low back pain (SEV) on disability (DIS) in 209 patients from
Suppose that, for each of three age groups (25-34, 35-44, and 45-54), we have recorded yearly sex-specific lung cancer mortality rates for the five-year period 19! through 1994. These data are to be
A five-year follow-up study was carried out to assess the relationship of diet and weight to the incidence of stomach cancer in 40- to 50-year-old males in a certain metropolitan area. Let K- denote
The following set of questions relates to using Poisson regression methods to analyze data from an in vitro study of human chromosome damage. In this study, using Poisson regression is appropriate
For the nonmelanoma skin cancer data considered in this chapter, consider the following additional model that was fit to these data: Model 5: In λij = a + 0 In Ti + βE + δE(In Ti) where Ti{is
The following questions concern Poisson regression models fit to fictitious follow-up study data in which rates of disease are modeled as a function of age and smoking status. The SAS program codes
A five-year follow-up study was carried out in a certain metropolitan area to assess the relationship of diet and weight to the incidence of stomach cancer. Data were obtained on n = 2,000 subjects.
Consider the following four correlation matrices that apply to clusters having five responses per cluster:a.b. c. d. a. Matrix A is an example of what kind of correlation structure? b. Matrix B is an
Consider the following four covariance matrices that apply to clusters having four responses per cluster:PQ R S a. Which of the above matrices are heterogeneous (as compared to homogeneous)
Consider again the study described in the main body of this chapter concerning the effect of an air pollution episode on pulmonary function measurements (FEV1) taken on each of K= 40 school children
This problem considers the (fictitious) study designed to compare two treatments for the relief of heartburn, described by the data set of Table 25.6. One of two treatments (A = active = 1, P =
Consider a different (fictitious) study to compare two treatments for the relief of heartburn. This new study involves a two-period "cross-over" design in which each subject is given two
The data set for this problem derives from the posture measurement study described in the main body of this chapter. Here we consider the data on shoulder flexion (SF) for 19 subjects that were each
The analysis described in Problem 1 for the posture measurement data on Shoulder Flexion (SF) may be criticized because information is lost when the 4 observations for a given subject on a given day
A study by Holder, Plikaytis, and Carlone (1996) compared two laboratory protocols designed to measure antibody levels in an enzyme-linked immunosorbent assay (ELISA) for Streptococcus pneumoniae.
A study by Heffner, Drawbaugh, and Zigmond (1974) investigated the effects of an amphetamine on the behavior of rats. Before the study began, 24 "thirsty" rats were trained to press a lever to obtain
Consider the same study by Heffner et al. (1974) and the data set described in Problem 4, in which the response measured was the lever press rate (LPR) used by a thirsty rat to press a lever and
Consider the same study by Heffner et al. (1974) and the data set described in Problems 4 and 5, where the response measured was the lever press rate (LPR) taken by a thirsty rat to press a lever and
A study by Rikkers et al. (1978) involved a prospective randomized surgical trial that compared cirrhotic patients who had bled from use of either a nonselective shunt (a standard operation) or a
In this problem, we consider the analysis of the combined information from both raters on the shoulder flexion (SF) scores in the posture measurement study. Thus, the questions below concern the data
In the problems below, assume that a significance level, a, of 0.05 and a power, (1 — /3), of 0.8 are desired unless otherwise stated. A clinical trial is to be conducted to investigate and compare
For the scenario in Problem 1(a), a. Determine the sample size for power levels ranging from 0.6 to 0.9, in increments of 0.05. Plot the sample size versus power. Comment on the observed
For the scenario in Problem 1(b), a. Determine the required sample size for power levels ranging from 0.6 to 0.9, in increments of 0.10. Plot the sample size versus power. Comment on the observed
In Example 27.3, the manufacturer of the drugs would like to determine how much more powerful the hypothesis-testing method would be if larger samples were collected. Plot the sample size versus the
A study is being planned to investigate the association between systolic blood pressure (SBP) and independent variables age (AGE), smoking history (SMK = 0 if non-smoker, SMK = 1 if a current or
Researchers in Example 27.9 feel that at most 200 patients can be enrolled in the study for each type of surgery. Redo the sample size calculations for that example and identify at least one
Redo Examples 27.11, 27.13, and 27.15 to determine the approximate sample size if a. The desired power is 0.9 (all other parameters are as given in the examples). b. The desired significance level is
A graduating engineer has signed up for three job interviews. She intends to categorize each one as being either a “success” or a “failure” depending on whether it leads to a plant trip.
Two darts are thrown at the following target:(a) Let (u, v) denote the outcome that the first dart lands in region u and the second dart, in region v. List the sample space of (u,
A woman has her purse snatched by two teenagers. She is subsequently shown a police lineup consisting of five suspects, including the two perpetrators. What is the sample space associated with the
In the game of craps, the person rolling the dice (the shooter) wins outright if his first toss is a 7 or an 11. If his first toss is a 2, 3, or 12, he loses outright. If his first roll is something
A probability-minded despot offers a convicted murderer a final chance to gain his release. The prisoner is given twenty chips, ten white and ten black. All twenty are to be placed into two urns,
Suppose that ten chips, numbered 1 through 10, are put into an urn at one minute to midnight, and chip number 1 is quickly removed. At one-half minute to midnight, chips numbered 11 through 20 are
Sketch the regions in the xy-plane corresponding to A ∪ B and A ∩ B if A = {(x, y): 0
Referring to Example 2.2.7, find A∩ B and A∪ B if the two equations were replaced by inequalities: x2 +2x ≤ 8 and x2 +x ≤6.
An electronic system has four components divided into two pairs. The two components of each pair are wired in parallel; the two pairs are wired in series. Let Ai j denote the event ith
Three dice are tossed, one red, one blue, and one green. What outcomes make up the event A that the sum of the three faces showing equals 5?
Define A = {x : 0 ≤ x ≤ 1}, B = {x : 0 ≤ x ≤ 3}, and C = {x : −1 ≤ x ≤ 2}. Draw diagrams showing each of the following sets of points: (a) AC ∩ B ∩C (b) AC ∪ (B ∩C) (c) A ∩ B
Suppose that each of the twelve letters in the wordis written on a chip. Define the events F, R, and C as follows:F: letters in first half of alphabetR: letters that are repeatedV: letters that are
Let A, B, and C be any three events defined on a sample space S. Show that (a) The outcomes in A ∪ (B ∩ C) are the same as the outcomes in (A ∪ B) ∩ (A ∪C). (b) The outcomes in A ∩ (B ∪
Let A1, A2. Ak be any set of events defined on a sample space S. What outcomes belong to the event(A1 ∪ A2 ∪· · ·∪ Ak)∪ (AC1 ∩ AC2 ∩· · ·∩ ACk )
Let A, B, and C be any three events defined on a sample space S. Show that the operations of union and intersection are associative by proving that (a) A ∪ (B ∪C) = (A ∪ B) ∪C = A ∪ B
Suppose that three events—A, B, and C—are defined on a sample space S. Use the union, intersection, and complement operations to represent each of the following events: (a) None of the three
Let events A and B and sample space S be defined as the following intervals: S ={x : 0≤ x ≤10} A ={x : 0
Pictured on the next page are two organizational charts describing the way upper management vets new proposals. For both models, three vice presidents€”1, 2, and 3€”each voice an
During orientation week, the latest Spiderman movie was shown twice at State University. Among the entering class of 6000 freshmen, 850 went to see it the first time, 690 the second time, while 4700
Let A and B be any two events. Use Venn diagrams to show that (a) The complement of their intersection is the union of their complements: (A ∩ B)C = AC ∪ BC (b) The complement of their union is
Let A, B, and C be any three events. Use Venn diagrams to show that (a) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (b) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Let A, B, and C be any three events. Use Venn diagrams to show that (a) A ∪ (B ∪ C) = (A ∪ B) ∪C (b) A ∩ (B ∩ C) = (A ∩ B) ∩C
Use Venn diagrams to suggest an equivalent way of representing the following events: (a) (A ∩ BC) C (b) B ∪ (A ∪ B) C (c) A ∩ (A ∩ B) C
A total of twelve hundred graduates of State Tech have gotten into medical school in the past several years. Of that number, one thousand earned scores of twenty-seven or higher on the MCAT and four
Let A, B, and C be any three events defined on a sample space S. Let N (A), N (B), N (C), N (A ∩ B), N (A ∩ C), N (B ∩ C), and N (A ∩ B ∩ C) denote the numbers of outcomes in all the
A poll conducted by a potential presidential candidate asked two questions: (1) Do you support the candidate’s position on taxes? and (2) Do you support the candidate’s position on homeland
Suppose that two cards are dealt from a standard 52-card poker deck. Let A be the event that the sum of the two cards is 8 (assume that aces have a numerical value of 1). How many outcomes are in A?
For two events A and B defined on a sample space S, N(A ∩ BC ) = 15, N(AC ∩ B) = 50, and N(A ∩ B) = 2. Given that N(S) = 120, how many outcomes belong to neither A nor B?
In the lingo of craps-shooters (where two dice are tossed and the underlying sample space is the matrix pictured in Figure 2.2.1) is the phrase “making a hard eight.” What might that mean?
A poker deck consists of fifty-two cards, representing thirteen denominations (2 through ace) and four suits (diamonds, hearts, clubs, and spades). A five-card hand is called a flush if all five
A telemarketer is planning to set up a phone bank to bilk widows with a Ponzi scheme. His past experience (prior to his most recent incarceration) suggests that each phone will be in use half the
According to a family-oriented lobbying group, there is too much crude language and violence on television. Forty-two percent of the programs they screened had language they found offensive, 27% were
An urn contains twenty-four chips, numbered 1 through 24. One is drawn at random. Let A be the event that the number is divisible by 2 and let B be the event that the number is divisible by 3. Find P
If State’s football team has a 10% chance of winning Saturday’s game, a 30% chance of winning two weeks from now, and a 65% chance of losing both games, what are their chances of winning exactly
Events A1 and A2 are such that A1 ∪ A2 = S and A1 ∩ A2 =∅. Find p2 if P (A1) = p1, P (A2) = p2, and 3p1 − p2 = 1 / 2.
Consolidated Industries has come under considerable pressure to eliminate its seemingly discriminatory hiring practices. Company officials have agreed that during the next five years, 60% of their
Three events—A, B, and C—are defined on a sample space, S. Given that P(A) = 0.2, P(B) = 0.1, and P(C)=0.3, what is the smallest possible value for P[(A ∪ B ∪C) C ]?
A coin is to be tossed four times. Define events X and Y such that X: first and last coins have opposite faces Y : exactly two heads appear Assume that each of the sixteen head/tail sequences has the
Two dice are tossed. Assume that each possible outcome has a 1 36 probability. Let A be the event that the sum of the faces showing is 6, and let B be the event that the face showing on one die is
Lucy is currently running two dot-com scams out of a bogus chat room. She estimates that the chances of the first one leading to her arrest are one in ten; the “risk” associated with the second
Let A and B be any two events defined on S. Suppose that P (A) = 0.4, P(B) = 0.5, and P (A ∩ B) = 0.1. What is the probability that A or B but not both occur?
Let A and B be two events defined on S. If the probability that at least one of them occurs is 0.3 and the probability that A occurs but B does not occur is 0.1, what is P (B)?
Suppose that three fair dice are tossed. Let Ai be the event that a 6 shows on the ith die, i = 1, 2, 3. Does P (A1 ∪ A2 ∪ A3) = 1 / 2? Explain.
Events A and B are defined on a sample space S such that P ((A ∪ B) C) = 0.5 and P (A ∩ B) = 0.2. What is the probability that either A or B but not both will occur?
Let A1, A2. An be a series of events for which Ai ∩ Aj = ∅ if i = j and A1 ∪ A2 ∪ · · · ∪ An = S. Let B be any event defined on S. Express B as a union of intersections.
Draw the Venn diagrams that would correspond to the equations (a) P(A ∩ B) = P(B) (b) P(A ∪ B) = P(B).
Suppose that two fair dice are tossed. What is the probability that the sum equals 10 given that it exceeds 8?
Suppose events A and B are such that P(A ∩ B)= 0.1 and P((A∪ B)C )=0.3. If P(A)=0.2, what does P[(A∩ B)|(A ∪ B)C ] equal? (Hint: Draw the Venn diagram.)
One hundred voters were asked their opinions of two candidates, A and B, running for mayor. Their responses to three questions are summarized below:(a) What is the probability that someone likes
A fair coin is tossed three times. What is the probability that at least two heads will occur given that at most two heads have occurred?
Two fair dice are rolled. What is the probability that the number on the first die was at least as large as 4 given that the sum of the two dice was 8?
Four cards are dealt from a standard 52-card poker deck. What is the probability that all four are aces given that at least three are aces? (Note: There are 270,725 different sets of four cards that
Given that P(A∩ BC)=0.3, P((A∪ B) C )=0.2, and P(A ∩ B)=0.1, find P(A|B).
Given that P(A)+ P(B)=0.9, P(A|B)=0.5, and P(B|A)=0.4, find P(A).
Let A and B be two events defined on a sample space S such that P(A ∩ BC) = 0.1, P(AC ∩ B) = 0.3, and P((A ∪ B) C )=0.2. Find the probability that at least one of the two events occurs given
Suppose two dice are rolled. Assume that each possible outcome has probability 1/36. Let A be the event that the sum of the two dice is greater than or equal to 8, and let B be the event that at
According to your neighborhood bookie, five horses are scheduled to run in the third race at the local track, and handicappers have assigned them the following probabilities of winning:Suppose that
Find P(A ∩ B) if P(A) = 0.2, P(B) = 0.4, and P(A|B)+ P(B|A)=0.75.
Andy, Bob, and Charley have all been serving time for grand theft auto. According to prison scuttlebutt, the warden plans to release two of the three next week. They all have identical records, so
An urn contains six white chips, four black chips, and five red chips. Five chips are drawn out, one at a time and without replacement. What is the probability of getting the sequence (black, black,
A man has n keys on a key ring, one of which opens the door to his apartment. Having celebrated a bit too much one evening, he returns home only to find himself unable to distinguish one key from
A toy manufacturer buys ball bearings from three different suppliers—50% of her total order comes from supplier 1, 30% from supplier 2, and the rest from supplier 3. Past experience has shown that
A fair coin is tossed. If a head turns up, a fair die is tossed; if a tail turns up, two fair dice are tossed. What is the probability that the face (or the sum of the faces) showing on the die (or
Foreign policy experts estimate that the probability is 0.65 that war will break out next year between two Middle East countries if either side significantly escalates its terrorist activities.
A telephone solicitor is responsible for canvassing three suburbs. In the past, 60% of the completed calls to Belle Meade have resulted in contributions, compared to 55% for Oak Hill and 35% for
If men constitute 47% of the population and tell the truth 78% of the time, while women tell the truth 63% of the time, what is the probability that a person selected at random will answer a question
If P (A|B)< P (A), show that P (B|A)< P (B).
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