# MATHEMATICS

23. A study of the salaries of full professors at a small university shows that the median salary for female professors is considerably less than the median male salary. Further investigation shows that the median salaries for male and female full professors are about the same in every department (English, physics, etc.) of the university. Which phenomenon explains the reversal in this example? A) extrapolation B) Simpson’s paradox C) causation D) correlation

Questions 24 – 26 Is age a good predictor of salary for CEO’s? Sixty CEO’s between the age of 32 and 74 were asked their salary (in thousands). The results of a statistical analysis are shown below: Simple linear regression results: Dependent Variable: SALARY Independent Variable: AGE SALARY = 242.70212 + 3.1327114 AGE Sample size: 59 R (correlation coefficient) = 0.1276 R-sq = 0.016270384 Estimate of error standard deviation: 220.64246 Parameter estimates: Parameter Estimate Std. Err. Alternative DF T-Stat P-Value Intercept 242.70212 168.7604 ≠ 0 57 1.4381461 0.1559 Slope 3.1327114 3.2264276 ≠ 0 57 0.9709536 0.3357 Analysis of variance table for regression model: Source DF SS MS F-stat P-value Model 1 45896.027 45896.027 0.9427509 0.3357 Error 57 2774936.2 48683.094 Total 58 2820832.2 24. Suppose a CEO is 57 years old. What do you predict his/her salary to be?

A) over $400,000 B) between $100,00 and $400,000 C) under $100,00 D) None of the above.

25. Suppose you wanted to predict the salary of the CEO of Facebook, Mark Zuckerberg, based on the

information here. How well do you think your prediction would be assuming Mr. Zuckerberg was 23 when he started Facebook and became CEO? A) The prediction would be accurate and around $300,000. B) The prediction would require extrapolation and therefore would not be accurate. C) The prediction would be accurate and around $240,000. D) None of the above.

26. What are possible reasons for a correlation around 0.13 for the above data?

A) Age is a very strong predictor of CEO salary. B) Age is not a good predictor and something else may be a better a predictor C) There is not enough data to accurately estimate the correlation. D) The range of ages is too small.

0.0 0.5 1.0 1.5 2.0 2.5

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27. Consider the scatterplot at right. What do we call the point indicated by the plotting symbol O? A) a residual B) influential C) a z-score

Questions 28 – 29 The 94 students in a statistics class are categorized by gender and by the year in school. The numbers obtained are displayed below:

Year in school Gender Freshman Sophomore Junior Senior Graduate Total Male 1 2 9 17 2 31 Female 23 17 13 7 3 63 Total 24 19 22 24 5 94

28. What proportion of the statistics students in this class are sophomores, given they are female?

A) 0.11 B) 0.202 C) 0.27 D) 19 29. What proportion of the statistics students in this class are male?

A) 0.065 B) 0.105 C) 0.33 D) 31 30. What is the best way to control for lurking variables?

A) Compare two or more treatments B) Randomize to assign experimental units to treatments C) Repeat each treatment on many units D) None of the above

Extra Credit 1. In order to determine if drinking from plastic water bottles causes cancer, researchers surveyed a large

sample of adults. For each adult they recorded whether the person drank regularly from plastic water bottles at any period in their life and whether the person had cancer. They then compared the proportion of cancer cases in those who drank from plastic water bottles regularly at some time in their lives with the proportion of cases in those who never drank from plastic water bottles at any point in their lives. The researchers found a higher proportion of cancer cases among those who drank from plastic water bottles regularly than among those who never drank from plastic water bottles. What type of study is this? A) An observational study B) An experiment but not a double-blind experiment C) A double-blind experiment D) A block design

2.. Which of the following best describes a simple random sample (SRS) of size n?

A) It is a random sample of size n selected so that everyone in the population has a known probability of being included in the sample.

B) It is a random sample of size n selected so that everyone in the population has the same chance of being included in the sample.

C) It is a probability sample of size n with known probabilities of selection. D) It is a sample selected from the population in such a way that every set of n individuals has an

equal chance of being in the sample actually selected. E) It is a sample of n individuals selected in such a way that only chance determines who is included

in the sample.

A common fear for incoming freshman in college is the dreaded “freshman fifteen.” The combination of being in a new environment away from home, a high stress level, alcohol consumption, and eating dining hall food can cause weight gain in college students. A study examined weight gained during the first year of college and what factors contribute to it. A 27-question survey was sent to 252 students at over 50 universities in the United States. Questions included information on demographics, weight gain, diet, family relationships, etc. Ninety-five survey responses were received from students across 37 United States colleges and universities, with 32 respondents from Rose-Hulman Institute of Technology.

3. What is the sample in this study?

A) U.S. college students B) All college students C) The survey respondents D) The 50 universities

4. What is the response rate?

A) 50/252 B) 95/252 C) 32/50 D) 32/25

5. What type of sample is this?

A) Simple random sample B) Probability sample C) Stratified random sample D) Voluntary response sample

6. Does the survey suffer from nonresponse?

A) No, everyone chosen for the survey participated. B) No, this was an experiment so nonresponse is not an issue. C) Yes, because not everyone chosen for the survey participated. D) Yes, because the survey contained too many questions and it is likely participants did not answer all the questions.

7. What could you do to improve the study?

A) Increase the coverage of universities that were selected for the study. B) Conduct a matched-pairs design instead. Weigh students on the first day of class and at the end

of their freshman year. C) Follow up with students who did not respond to the study to improve the response rate. D) All of the above could be done to improve the study.

8. One of the questions asked was, “How much weight did you gain after your freshman year?” This

would be an example of ______. A) poor wording of a question because some students may have lost weight. B) response bias because the question does not allow for a valid response from students who lost

weight. C) voluntary response because the students can write whatever they want. D) only A and B. E) None of the above

Chs 4 – 5 1. A penny is tossed. We observe whether it lands heads up or tails up. Suppose the penny is a fair

coin, i.e., the probability of heads is ½ and the probability of tails is ½. What does this mean? A) Every occurrence of a head must be balanced by a tail in one of the next two or three tosses. B) If the coin is tossed many, many times, the proportion of tosses that land heads will be

approximately ½, and this proportion will tend to get closer and closer to ½ as the number of tosses increases.

C) Regardless of the number of flips, half will be heads and half tails. D) All of the above.

2. Which of the following is (are) appropriate statements about randomness and/or probability?

A) A phenomenon is called random if individual outcomes are uncertain, but in a large number of repetitions, there is a regular distribution of outcomes.

B) The word random in statistics is a description of a kind of order that emerges in the long run. C) Probability describes only what happens in the long run. D) In a small or moderate number of repetitions, the observed proportion of an outcome can be far

from the probability of the outcome. E) All of the above are appropriate statements.

3. Suppose a fair coin is flipped twice and the number of heads is counted. Which of the following is a

valid probability model for the number of heads observed in two flips? A) Number of heads 0 1 2 C) Number of heads 0 1 2

Probability ¼ ½ ½ Probability ¼ ¼ ¼

B) Number of heads 0 1 2 D) None of the above. Probability ⅓ ½ ⅓

Use the following to answer questions 4–5: Ignoring twins and other multiple births, assume babies born at a hospital are independent events with the probability that a baby is a boy and the probability that a baby is a girl both equal to 0.5. 4. What is the probability that the next three babies are of the same sex?

A) 0.125 C) 0.250 B) 0.375 D) 0.500

5. Define event B = {at least one of the next two babies is a boy}. What is the probability of the

complement of event B? A) 0.125 C) 0.250 B) 0.375 D) 0.500

6. The American Veterinary Association claims that the annual cost of medical care for dogs averages

$100 with a standard deviation of $30. The cost for cats averages $120 with a standard deviation of $35. Some basic algebraic and statistical steps show us that the average of the difference in the cost of medical care for dogs and cats is then $100 –$120 = –$20. The standard deviation of that same difference equals $46. If the difference in costs follows a Normal distribution, what is the probability that the cost for someone’s dog is higher than for the cat? A) 0.2839 C) 0.6618 B) 0.3319 D) 0.7161

Use the following situation to answer questions 7–8: A study was conducted in a large population of adults concerning eyeglasses for correcting reading vision. Based on an examination by a qualified professional, the individuals were judged as to whether or not they needed to wear glasses for reading. In addition it was determined whether or not they were currently using glasses for reading. The following table provides the proportions found in the study: Used glasses for reading Yes No Judged to need Yes 0.42 0.18 glasses No 0.04 0.36 7. If a single adult is selected at random from this large population, what is the probability that the adult is

judged to need eyeglasses for reading? A) 0.46 C) 0.78 B) 0.42 D) 0.60

8. What is the probability that the selected adult is judged to need eyeglasses but does not use them for

reading? A) 0.42 C) 0.54 B) 0.18 D) 0.60

9. Consider the following probability distribution for a discrete random variable X:

X 3 4 5 6 7 P(X = x) 0.15 0.10 0.20 0.25 0.3