# Solutions - Kenyon College

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Elements of Statistics (Math 106) - Exam 1
Spring 2009 - Brad Hartlaub
Directions: Please answer all of the questions below and show your work. The point values for each
problem are indicated in parentheses. You may use one sheet of formulas, our course web page, and
Minitab during the exam. Don't spend too much time on anyone part of the exam.
1. Thirty percent of all automobiles undergoing an emissions inspection at a certain inspection station
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a. Among 15 randomly selected cars, what is the probability that at most 5 fail the inspection? (5)
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b. Among 15 randomly selected cars, what is the probability that exactly 5 fail the inspection? (5)
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Among 25 randomly selected cars, what is the mean value of the number of cafs&middot;that
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2. What type of graph or graphs would you plan to make in a study of each of the following issues?
(18 - 3 each question)
a. What makes of cars do students drive?
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How many hours per week do stGdents study?
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3. You are assigned to direct a study at Kenyon College to discover factors that are associated with
strong academic performance. You decide to identify 20 students who have perfect GP As of 4.0, and
then measure explanatory variables for them that you think may be important, such as high school
GPA and average amount oftime spent studying per day.
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b. Describe a study design that would provide more useful information. (10)
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groups representing the relative sizes of the groups in the population. For instance, the quota might be
50 factory workers, 100 housewives, 60 elderly people, 30 Hispanics, and so forth. This is called
quota sampling. Is this a random sampling method? Explain, and discuss the potential advantages
and disadvantages of this method. (15)
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5. Raw scores on behavioral tests are often transformed for easier comparison. A test of reading
ability has mean 75 and standard deviation 10 when given to third graders. Sixth graders have mean
score 82 and standard deviation lion the same test. To provide separate &quot;norms&quot; for each grade, we
want scores in each grade to have mean 100 and standard deviation 20.
a. What linear transformation will change third-grade scores x into new scores
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= a + bx that
have the desired mean and standard deviation? (Use b &gt; 0 to preserve the order of the scores.)
(10)
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b. The linear transformation that will change sixth-grade scores x mto new scores that have the
desired mean and standard deviation is xnew = -49.0924 + 1.8182x. Nancy is a sixth-grade
student who scores 78 on the test. What is her transformed score? David is a third-grade
student who scores 78. Who scores higher within his or her grade? (10)
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6. The usual way to study the brain's response to sounds is to have subjects listen to &quot;pure tones.&quot;
The response to recognizable sounds may differ. To compare responses, researchers anesthetized
macaque monkeys. They fed pure tones and also monkey calls directly to their brains by inserting
electrodes. Response to the stimulus was measured by firing rate (electrical spikes per second) of
neurons in various areas of the brain. The file p:\data\math\stats\monkey.mtw contains the responses
for 37 neurons.
a. One notable finding is that responses to monkey calls are generally stronger than responses to
pure tones. Give a numerical
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monkey call response when the pure tone response is 550? Explain. (5)
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7. Different varieties of the tropical flower Heliconia are fertilized by different species of
hummingbirds. Over time, the lengths of the flowers and the form of the hummingbirds' beaks have
evolved to match each other. The data on the lengths in millimeters of three varieties of these flowers
on the island of Dominca are in the file p:\data\math\stats\heliconia.mtw.
Use visual displays and
descriptive statistics to compare the three distributions. Your comparison should address center,
spread, and shape of the three distributions. What are the most important differences among the three
varieties of flowers? (30)
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