Q1. 1a. Three draws will be made at random with replacement from the box below. How many possible ordered ways are there for the three draws to turn out?
The following pertain to questions #1b, #1c, and #1c.
One ticket will be drawn at random from each of the two boxes, "A" and "B", shown below
1b. What is the probability that the number drawn from A equals the number drawn from B?
1c. What is the probability that the number drawn from A is larger than the number drawn from B?
1d. What is the probability that the number drawn from A is smaller than the number drawn from B?
1e. Two draws are made at random with replacement from the box below. What is the probability that the sum of the draws will equal 4?
Q2. 2a. Consider the following newspaper article, quoting a consultant in the field of fertility treatment:
"At his clinic, he says, on the first cycle of fertility treatment, 1/3 get pregnant, 1/3 do not and drop out, and 1/3 try again. Of those who go for a second treatment, there is a similar split of 1/3 successful, 1/3 drop out and 1/3 try again. The consultant claims that those who persist have an increased chance each time, on the basis that the pool is smaller and yet a third must get pregnant".
Is the consultant's reasoning correct? Use your understanding of basic probabilities to explain your answer.
2b. A computing magazine reported that 1/3 of graduates working in computing have degrees in information technology (IT) and 2/3 have degrees in other subjects. They conclude that:
"arts and science graduates are twice as likely as IT graduates to pursue careers as IT professionals".
In 1-2 sentences, explain why this conclusion is incorrect.
Q3. The following table is from a case-control study into maternal smoking during pregnancy and Down syndrome. It shows the basic characteristics of mothers giving birth to babies with Down syndrome (cases), and without Down syndrome (controls).
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Cases (n=775)
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Controls (n=7750)
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Number
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%
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Number
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%
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Smoking during pregnancy, among
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Age < 35 years
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|
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Yes
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112
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20.0
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1411
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20.2
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No
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421
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75.0
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5214
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74.6
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|
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Unknown
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28
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5.0
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363
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5.2
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Age ≥ years
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|
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Yes
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15
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7.0
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108
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14.2
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|
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No
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186
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86.9
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611
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80.2
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Unknown
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13
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6.1
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43
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5.6
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3a. Use the information in the table to construct a 2x2 table for women age < 35. For purposes of this exercise, exclude from analysis any woman for whom smoking status is Unknown.
3b. Use the information in the table to construct a 2x2 table for women age > 35. Again, for purposes of this exercise, exclude from analysis any woman for whom smoking status is Unknown.
3c. Consider your 2x2 table for women age < 35. For these women, among mothers giving birth to babies with Down syndrome, by hand, calculate the odds that they had smoked during pregnancy. Again for these women, among mothers giving birth to a healthy baby, calculate by hand the odds that they had smoked during pregnancy. Finally, use your two odds calculations to obtain the odds ratio. In 1-2 sentences, explain the odds ratio in lay terms.
3d. Repeat the calculations you did to answer question 3c, this time for women age > 35. In 1¬2 sentences, explain the odds ratio in lay terms.
3e. Finally, in 1-2 more sentences, what is your interpretation of the comparison of the two odds ratios? That is, what does this comparison suggest about age as a risk factor for delivering a Down syndrome baby?
Q4. In a certain study, a sample of persons remanded into custody (incarcerated) in a certain jail were given a detailed interview. From this interview, researchers were able to ascertain who did and who did not take drugs at the time of incarceration. The following table shows the research findings for cannabis and it shows whether the prison's screening procedure detected cannabis use. For this question, assume that the research findings are truth.
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Research Finding
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Prison Screening Result
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User
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Non-user
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User
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49
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6
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Non-user
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201
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118
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4a. Treating the research findings as truth, by hand, calculate the sensitivity of the prison screening procedure. Show your work.
4b. Treating the research findings as truth, by hand, calculate the specificity of the prison screening procedure. Show your work.
4c. Treating the research findings as truth, by hand, calculate the predictive value positive. Show your work.
4d. Treating the research findings as truth, by hand, calculate the predictive value negative. Show your work.
4e. Calculate the overall probability of concordance (Recall: concordance refers to both procedures, yielding the same result). Show your work.
Q5. The prevalence of undetected diabetes in a certain population is 1.5%. A new screening test measures blood serum glucose content. A value of 180 mg% or higher is considered positive. The sensitivity and specificity associated with this screening test are 22.9% and 99.8% respectively.
By hand, calculate the predictive value of a positive test. Show your work.
Q6. 6a. A study was conducted to relate the intelligence of 18-year old men to the number of their brothers and sisters. In this particular country, all men take a military pre-induction examination at age 18. The exam includes an intelligence test known as "Raven's progressive matrices" and includes questions about demographic variables like family size. The study related scores on the intelligence test with family size.
i) What is the population?
ii) What is the sample?
6b. In one study, the Educational Testing Service needed a representative sample of college students. To draw the sample, they first divided up the population of all colleges and university into relatively homogeneous groups. One group consisted of all public universities with 25,000 or more students. Another group consisted of all private four-year colleges with 1000 or fewer students: and so on. Then they used their judgment to choose one representative school from each group. That created a sample of schools. Each school in the sample was then asked to pick a sample of students.
i) Was this a good way to get a representative sample of students?
ii) What approach would you have used?
6c. The monthly Gallup poll opinion survey is based on a sample of about 1,500 persons "scientifically chosen as a representative cross section of the American public". The Gallup poll thinks the sample is representative mainly because (CHOOSE ONE):
i) It resembles the population with respect to such characteristics as race, sex, age, income, and education; or
ii) It was chosen using as probability method.
6d. A survey is carried out at a university to estimate the percentage of undergraduates living at home during the current semester. The registrar keeps an alphabetical list of all undergraduates, with their current addresses. Suppose there are 10,000 undergraduates in the current semester. Someone proposes to choose a number at random from one to a hundred, count that far down the list, taking thdt name and every hundredth name after it for the sample.
i) Is this a probability method?
ii) Is it the same as simple random sampling?
Q7. 7a. TRUE or FALSE: The number of sixes in 20 throws of a die follows a binomial distribution.
7b. TRUE or FALSE: The weight of a human follows a binomial distribution.
7c. TRUE or FALSE: The number of a random sample of patients who respond to a certain treatment follows a binomial distribution.
7d. TRUE or FALSE: The number of red cells in I ml of blood follows a binomial distribution.
7e. TRUE or FALSE: The proportion of hypertensives in a random sample of adult men follows a binomial distribution.
7f. TRUE or FALSE: If a coin is tossed twice in succession, the expected number of tails is 1.5.
7g. TRUE or FALSE: If a coin is tossed twice in succession, the probability of two tails is 0.25.
7h. TRUE or FALSE: If a coin is tossed twice in succession, the number of tails follows a Binomial distribution.
7i. TRUE or FALSE: If a coin is tossed twice in succession, the probability of at least one tail is 0.5.
7j. TRUE or FALSE: If a coin is tossed twice in succession, the distribution of the number of tails is symmetrical.
Q8. 8a. A citrus farmer has observed the following probability distribution for the number of oranges per tree. How many oranges does he/she expect on average? (Hint: This question is asking you to calculate a statistical expectation)
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Outcome, # oranges
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25
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30
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35
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40
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Probability
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.10
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.40
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.30
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.20
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8b. Consider again the probability distribution in question 8a. By hand, calculate the standard deviation of this distribution. Show your work.
8c. A manufacturer ships toasters in cartons of 20. In each carton, they estimate that each toaster has a 5% chance of being be malfunctioning, thus requiring it to be sent back for repairs. What is the probability that in one carton, there will be exactly 3 toasters that need repair?
8d. A soccer team estimates that they will score on 8% of the corner kicks. In next week's game, the team hopes to kick 15 corner kicks. What are the chances that they will score on exactly two of these opportunities?
8e. Consider again the setting in question 8d. If this soccer team has 200 corner kicks over the season, what are the chances that they will score more than 22 times?
Q9. 9a. The 68-95-99.7 rule is used as a guide when working with normally distributed random variables. Recall from the Unit 7 notes, this rule says that
68% of the distribution lies in the region μ ± σ
95% of the distribution lies in the region μ ± 2σ
99.7% of the distribution lies in the region μ ± 3σ
Thus, although p and a may be different from one normal distribution to another, you can always depend on the 68-95-99.7 rule to predict the percentage of individuals who will fall in these ranges.
The Wechsler Adult Intelligence Scale (WAIS) is a commonly used intelligence test that is calibrated to produce a Normal distribution of scores with it =100 and c =15 for various age groups. Based on the 68-95-99.7 rule, we can say that
68% of the distribution lies in the range _______ to _______
95% of the distribution lies in the range _______ to _______
99.7% of the distribution lies in the range _______ to _______
9b. Suppose it is known that heights of women in the United States is modeled well using a Normal distribution with μ = 163.3 cm and σ = 6.5 cm
How tall does a woman have to be to be taller than 90% of women?
How short does a woman have to be to be shorter than 90% of women?
Q10. 10a. Suppose it is known that the weights of brains from Alzheimer cadavers is modeled well using a Normal distribution with μ = 1077 g and σ = 106 g. What proportion of cadaver brains with Alzheimer disease will weigh more than 1250 g?
10b. Suppose it is known that water samples from a particular site demonstrate a mean coliform level of 10 organisms per liter with standard deviation 2 organisms per liter. Suppose further that these values are modeled well using a Normal distribution. What percentage of samples will contain more than 15 organisms per liter?
10c. The SAT and ACT are standardized tests for college admission in the United States. Both tests include components that measure reading comprehension. Suppose that SAT critical reading scores are normally distributed with μ = 510 and σ = 115. Suppose further that ACT critical reading scores are normally distributed with μ = 20.5 and σ = 5.
Sam takes the SAT reading test and scores 660. Dave takes the ACT test and scores 28. Who had the superior score, Sam or Dave? Show your work.