Exam 1 Name
______________________________
STA 2023, Spring 2007
Instructions: Please show your work and clearly indicate
your answers. For full credit, your work
and/or explanation must support your answer.
Always specify units. Provide
explanations where requested!
I expect you to exhibit a level of individual academic integrity that is
commensurate with being a part of the
Please acknowledge that integrity by signing the honor statement at the end of the test.
1.
(18 points) In 2004, the Boston Red Sox won the World
Series for the first time in 86 years.
The table below gives the salaries of the Red Sox players as of opening
day of the 2005 season.
|
Player |
Salary |
|
Player |
Salary |
|
Ramirez |
$19,806,820 |
|
Embree |
$3,000,000 |
|
Schilling |
$14,500,000 |
|
Timlin |
$2,750,000 |
|
Damon |
$8,250,000 |
|
Bellhorn |
$2,750,000 |
|
Renteria |
$8,000,000 |
|
Mueller |
$2,500,000 |
|
Varitek |
$8,000,000 |
|
Arroyo |
$1,850,000 |
|
Nixon |
$7,500,000 |
|
Mirabelli |
$1,500,000 |
|
Foulke |
$7,500,000 |
|
Miller |
$1,500,000 |
|
Clement |
$6,500,000 |
|
Halama |
$850,000 |
|
Ortiz |
$5,250,000 |
|
Mantei |
$750,000 |
|
|
$4,670,000 |
|
Vazquez |
$700,000 |
|
Wells |
$4,075,000 |
|
Myers |
$600,000 |
|
Payton |
$3,500,000 |
|
Youkilis |
$323,125 |
|
Millar |
$3,500,000 |
|
Stern |
$316,000 |
a. Illustrate the distributions of
salaries with a histogram or stemplot. (If you use a histogram, say what your bins
include. If you use a stemplot, say what your stems and leaves represent. Label appropriately.)
b. Give an
appropriate numerical summary for the distribution.
c. Write a sentence or two describing the
distribution. Be sure to address
features of the distribution that were discussed repeatedly in class and in
your text.
d. Which members, if any, of the team
have salaries that are considered outliers by the 1.5 * IQR Rule? Your work should support your answer.
2.
(10 points) The figure below shows the number of
servings of fruit per day claimed by 74 seventeen-year-old girls in a study in
a. Describe
this distribution in words. Address the
features discussed in class.
b. What percent
of the girls ate fewer than two servings of fruit per day?
c. Compute the
5 number summary for these data.
3.
(12 points) Mechanical measurements on supposedly
identical objects usually vary. The
variation often follows a normal distribution.
The stress required to break a type of bolt varies Normally
with mean 75 kilopounds per square inch (ksi) and standard deviation 8.3 ksi.
a. Find the
z-score for a bolt that breaks at a stress of 90 ksi.
b. What proportion
of these bolts will withstand a stress of 90 ksi
without breaking?
c. What range
covers the middle 50% of breaking strengths for these bolts?
4.
(4 points)
a. Choose four numbers from the whole
numbers 0 to 10 (repeats allowed) with the smallest possible standard
deviation.
b. Choose four
numbers from the whole numbers 0 to 10 (repeats allowed) with the largest
possible standard deviation.
5.
(20 points) The table below gives the heights (in
inches) of 11 adult brother-sister pairs.
|
Brother |
71 |
68 |
66 |
67 |
70 |
71 |
70 |
73 |
72 |
65 |
66 |
|
Sister |
69 |
64 |
65 |
63 |
65 |
62 |
65 |
64 |
66 |
59 |
62 |
a. Find the LSR
line for predicting a sister’s height from a brother’s height. Record the equation of your line.
b. Damien is 70
inches tall. Predict the height of his
sister Tonya.
c. Use your
result from the previous part, along with another prediction (brother’s height
65 inches would be a good choice) to carefully
draw the LSR line on the scatterplot below.
d. One point on the scatterplot represents two observations. Circle it.
e. Based on the scatterplot
and the correlation r, do you expect
your prediction to be very accurate?
Explain, and include the value of r
in your response.
e. How would
the correlation change if all the men were 6 inches shorter than reported in
the table?
f. If heights
were measured in centimeters rather than inches, how would the correlation
change? (There are 2.54 cm in an inch.)
g. If every
sister was exactly 3 inches shorter than her brother, what would be the
correlation between brother and sister heights?
h. Report the
value of r2 and write a
sentence that carefully interprets this value in the context of sibling
heights.
6.
(18 points) We sometimes hear that getting married is
good for your career. The table below
presents data from one of the studies behind this generalization. To avoid gender effects, the investigators
looked only at men. The data describe
the marital status and job level of all 8235 male managers and professionals employed
by a large manufacturing firm. The job
grades are a measure of the value of a job to the company, with 1 being low and
4 being high.
|
|
Single |
Married |
Divorced |
Widowed |
Total |
|
Grade 1 |
58 |
874 |
15 |
8 |
955 |
|
Grade 2 |
222 |
3927 |
70 |
20 |
4239 |
|
Grade 3 |
50 |
2396 |
34 |
10 |
2490 |
|
Grade 4 |
7 |
533 |
7 |
4 |
551 |
|
Total |
337 |
7730 |
126 |
42 |
8235 |
a. Give (in percents – you may round to
the nearest full percent) the two marginal distributions (one for marital
status and one for job grade). You can
write your percentages at the margins of the table above if you’d like. Do each of your two
sets of percentages add to exactly 100%?
If not, why not?
b. Give (in percents) the conditional distribution of job grade among single
men.
c. Give (in
percents) the conditional distribution of job grade among married men.
d. Briefly explain
the relationship that your conditional distributions reveal.
e. We should
not conclude that single men can help their careers by getting married. What lurking variables might help explain the
association between marital status and job grade?
7.
(6 points) We
expect that students who do well on the midterm exam in a course will usually
also do well on the final exam.
Professor Smith looked at the exam scores of all 346 students who took
his statistics class over a 10 year period.
The least-square line for predicting the final exam score from
midterm-exam score was
.
Octavio scores 10 points above the class mean on the
midterm. How many points above the class
mean do you predict that he will score on the final? (Hint:
Use the fact that the LSR line passes through the point 
and the fact that Octavio’s midterms score is 
.)
8.
(12 points)
Wabash Tech has two professional schools, business and law. The tables below show applicants to both
schools, categorized by gender and admission decision, as well as the combined
totals
|
|
|
|
|
Totals |
||||||
|
|
Admit |
Deny |
|
|
Admit |
Deny |
|
|
Admit |
Deny |
|
Male |
480 |
120 |
|
Male |
10 |
90 |
|
Male |
490 |
210 |
|
Female |
180 |
20 |
|
Female |
100 |
200 |
|
Female |
280 |
220 |
a. Calculate
the percent of all male applicants that are admitted and the percent of all
female applicants that are admitted.
males __________ females
_________
b. Now compute separately the percents
of male and female applicants admitted by the business school and by the law
school.
Business males __________ females _________
Law males __________ females _________
c. What is
interesting about your answers to the previous parts? This phemonmenon
has a name… what is it?
d. Explain
carefully, as if speaking to a skeptical reporter, how it can happen that
I have abided by the principles of the Honor Code in completing this test. _________________________________________
Signature
Exam 2 Name
______________________________
STA 2023, Spring 2007
- (8 points)
The Ministry of Health in the Canadian
a. What is the population for this survey? What is the sample?
population:
sample:
b. The survey found that 76% of males and 86% of females in the sample had visited a general practitioner at
least once in the past year. These values are (circle one): parameters or statistics.
c. Do you think these estimates are close to the truth about the entire population? Why?
- (10 points) The level of nitrogen oxides in the
exhaust of a particular car model varies with mean 0.9 grams per mile
(g/mi) and standard deviation 0.15 g/mi.
A company has 125 cars of this model in its fleet.
a.
What is the
approximate distribution of the mean
for the company’s
cars?
b.
What is the
probability that the mean
c.
What is the level
L such that the probability that is greater than L is only 0.01?
- (14 points)
People who eat lots of fruits and vegetables have lower rates of
colon cancer than those who eat little of these foods. Fruits and vegetables are rich in
“antioxidants” such as vitamins A, C, and E. Will taking antioxidants help prevent
colon cancer? A medical experiment
studied this question with 864 people who were at risk of colon
cancer. The subjects were divided
into four groups: daily
beta-carotene, daily vitamins C and E, all three vitamins every day, or
daily placebo. After four years,
the researchers were surprised to find no significant difference in colon
cancer among the groups.
a. What are the explanatory and response variables in this study?
explanatory:
response:
b. Outline the design of the experiment, specifying group sizes and explicitly explaining how
randomization is used.
c. The study was double-blind. What does this mean?
d. What does “no significant difference” mean in describing the outcome of the study?
e. Suggest some lurking variables that could explain why people who eat lots of fruits and
vegetables have lower rates of colon cancer. The experiment suggests that these variables,
rather than the antioxidants, may be responsible for the observed benefits of fruits and
vegetables.
4. (4 points) A researcher looking for evidence of
extrasensory perception (
- (15 points)
A couple plans to have three children. There are 8 possible arrangements of
girls and boys. For example,
a. Write down all 8 arrangements of the sexes of the three children. What is the probability of any
one of these arrangements?
b. Let X be the number of girls the couple has. What is the probability that X = 2?
c. Find the distribution of the random variable X. That is, list the possible values X can take
along with the probability for each.
d. Ashley has three children. Given that one is a girl, what is the conditional probability that she has
three girls?
e. Brianna also has three children. Given that her oldest child is a girl, what is the conditional
probability that she has three girls?
6. (6 points)
a. When
asked to explain the meaning of “the P-value was P = 0.03,” a student says,
“This means
there is only probability 0.03 that
the null hypothesis is true.” Is this an
essentially correct
explanation? Explain your answer fully.
b. Another
student, when asked why statistical significance appears so often in research
reports,
says, “Because saying that results
are significant tells us that they cannot easily be explained by
chance variation alone.” Do you think that this statement is
essentially correct? Explain your
answer fully.
- (16
points) Leaking from underground gasoline tanks at service stations can
damage the environment. It is
estimated that 25% of these tanks leak.
You examine 15 tanks chosen at random, independently from one
another. Let X be the number of tanks (out of 15) that are leaking.
a.
Explain why you
expect X to follow a binomial
distribution. (You should verify four
conditions for the binomial setting.)
b.
What is the
probability that exactly 5 of the 15 tanks leak?
c.
Now you do a
larger study, examining a random sample of 900 tanks nationally. What is the mean and standard deviation for
the number of tanks in 900 that are leaking?
mean:
standard deviation:
d.
What is the
approximate probability that at least 300 of these 900 tanks are leaking?
e.
In part (b) you
found the probability that exactly 1/3 of the sample was leaking, while
in part (d) you found the probability that at least 1/3 of the sample
was leaking. Yet, the probability from
part (d) is (or should be!) much smaller than that from part (b). Explain why this makes sense.
- (12 points) A recent article in the Palm Beach
Post discussed the amounts of money spent by parents to prepare their
children to take the
a.
Find the z*
critical value that would be used to compute a 92% confidence interval for the
mean increase in
b.
After taking the
hypotheses:
test statistic:
p-value:
conclusion:
- (15 points) Here is a distribution of the lengths
in feet of 44 great white sharks, together with some summary
statistics. Assume the standard deviation
for the lengths of all great white sharks is s = 3 feet.
a.
Describe the
distribution. Is it reasonable to think
these data might have come from a normal population?
b. Give a 95% confidence interval for the mean length of
great white sharks. Give a one sentence
interpretation of your interval, in the context of sharks.
c. How large of a sample would be needed to estimate the
mean length of all great white sharks to within 0.5 feet?
d.
Based on the
interval you found above, is there significant evidence at the 5% level to
reject the claim that “Great white sharks average 20 feet in length”? Explain briefly.
- (BONUS
8 points) The unique colors of
cashmere sweaters your firm makes result from heating undyed
yearn in a kettle with a dye liquor. The pH (acidity) of the liquor is
critical for regulating dye uptake and hence the final color. There are 5 kettles, all of which
receive dye liquor from a common source.
Twice each day, the pH of the liquor in each kettle is measured,
giving a sample of size 5. The
process has been operating in control with m
= 4.22 and s = -0.127.
a. Give the center line and control limits for the chart.
b. What are the natural tolerances for the individual pH measurements?
c. Explain what an control chart is
used for.
I have adhered to the
principles of the Honor Code in completing this test. __________________________
Exam 3 Name
______________________________
STA 2023, Spring 2007
Show your work on all problems. Answers with no work will receive no
credit. When you are asked for
conclusions or interpretations of your results, your response should say
something about the original data.
Include units whenever appropriate.
If you use you calculator for
something other than arithmetic, say what you did with your calculator and
indicate what function(s) you used. You
are strongly encouraged to use your calculator to check the results of your
confidence intervals and tests for significance whenever possible (and you need
not write anything about these confirmations), but your work should exhibit
that you understand how to perform the required procedure.
I expect you to exhibit a level of individual academic
integrity that is commensurate with being a part of the
11. (14 points)
Some FAU students are concerned about having two final exams on the same
day. To investigate the extent of this
problem, the student government would like to conduct a survey.
a.
If the student government
wants to estimate the proportion of students who have two finals on the same
day to within 0.03 with 90% confidence, how many students should be surveyed?
b.
Suppose the
student in charge of the survey eventually uses a sample of size n = 40 and
finds that 6 students have two exams on the same day. Use the plus 4 method to construct an 96% confidence interval for the overall proportion of
students at FAU who have this problem.
Write a sentence interpreting your result.
12.
(16 points) A biologist who studies spiders is interested
in comparing the lengths of male and female green lynx spiders. Summary statistics and plots for the lengths,
in mm, of 30 male spiders and 25 female spiders are shown below.
a.
Describe the
distributions of the spider lengths.
Compare and contrast the male and female distributions.
b.
Do these data
meet our technical assumptions for inference?
Explain why or why not.
c.
Find a 95%
confidence interval for the difference in mean lengths of the two groups of
spiders. Write a one or two sentence
interpretation of your confidence interval.
13.
(16 points) The
design of controls and instruments affects how easily people can use them. A student project investigated this effect by
asking 15 right-handed students to turn a knob (with their right hands) that
moved an indicator by screw action.
There were two identical instruments, one with a right-hand thread (the
knob turns clockwise) and the other with a left-hand thread (the knob turns
counterclockwise). The table below gives
the time in seconds each subject took to move the indicator a fixed distance. Note
that each subject used both instruments!!!
|
Subject |
RightThread |
LeftThread |
Difference |
|
1 |
113 |
137 |
-24 |
|
2 |
105 |
105 |
0 |
|
3 |
130 |
133 |
-3 |
|
4 |
101 |
108 |
-7 |
|
5 |
138 |
115 |
23 |
|
6 |
118 |
170 |
-52 |
|
7 |
87 |
103 |
-16 |
|
8 |
116 |
145 |
-29 |
|
9 |
75 |
78 |
-3 |
|
10 |
96 |
107 |
-11 |
|
11 |
122 |
84 |
38 |
|
12 |
103 |
148 |
-45 |
|
13 |
116 |
147 |
-31 |
|
14 |
107 |
87 |
20 |
|
15 |
118 |
166 |
-48 |
a.
Each of the 15
students used both instruments. Discuss
briefly how you would use randomization in arranging the experiment.
b.
The project hoped
to show that right-handed people find right-hand threads easier to use. What is the parameter(s) for the appropriate
test? (Describe the parameter(s) in
words, and define the symbol(s) you will use in conducting a hypothesis test.)
c.
Conduct the
appropriate test. State your hypotheses,
compute the test statistic, give the P-value, and report your conclusions in
the context of the experiment.
14. (16 points)
Jeanie completed a thesis project on the effect of renourished
beaches on loggerhead turtle nesting.
She considered two stretches of beach, one that was renourished
between the 2001 and 2002 turtle nesting seasons, and one that was left
natural. One of the things Jeanne
studied was the “nesting success rate” of loggerhead turtles, which is found by
dividing the number of nests by the number of trips (called “crawls”) that
turtles make onto the beach, which may or may not result in a nest. In the 2001 nesting season, both stretches of
beaches that Jeannie considered had a nesting success rate of approximately
0.396, and this is known to be close to the historical average nesting success
rate in this geographic region. In 2002,
the renourished beach had 434 nests from 1339 crawls,
for a success rate of 0.3241.
a.
Is there evidence
that the nesting success rate of loggerhead turtles in 2002 was smaller than
the normal rate? State your hypotheses,
both in words and in symbols, report the test statistic and P-value and clearly
interpret your results in the context of turtle nesting.
b.
It might be the
case that 2002 was a less successful nesting season for all stretches of beach,
and had nothing to do with the beach renourishment. To decide, Jeannie compared the 2002 nesting
success rates on the renourished and natural
beaches. In 2002, the natural beach had
223 nests from 539 crawls. Does the data
provide evidence of decreased nesting success of loggerhead turtles on renourished beaches compared to natural beaches in
2002?
State the hypotheses for this test, both in words and in symbols.
Which of the following is the correct test statistic computation? Circle one.
z = 
OR z =
Find the P-value and clearly
interpret your results in the context of turtle nesting.
15. (20 points) A State Highway Patrol Department would like to
assess if the cause of an accident is related to the outcome of the
accident. The Department decides to
focus on accidents that occur along the major toll road that crosses the
state. A random sample of 250 accidents
reports over the past six months were obtained.
The accidents were cross-classified by primary cause of the accident and
by the outcome of the accident. A
portion of the data and the table of expected counts is
below. Note that there is no totals for the table of expected counts.
|
|
Death |
No Death |
Total |
|
Speeding |
21 |
41 |
62 |
|
Recklessness |
10 |
|
|
|
Drinking |
39 |
71 |
110 |
|
Other |
5 |
|
|
|
Total |
75 |
175 |
250 |
|
EXPECTED COUNTS |
Death |
No Death |
|
Speeding |
18.6 |
43.4 |
|
Recklessness |
|
28 |
|
Drinking |
33 |
77 |
|
Other |
|
26.6 |
a. Fill in the blank spaces in
the tables above.
b. Calculate the proportion of
accidents resulting in deaths for each of the causes. Based solely on this information, do you
think there is an association between cause and result? Why or why not?
|
|
Probability of death, given… |
|
Speeding |
|
|
Recklessness |
|
|
Drinking |
|
|
Other |
|
c. State the null and
alternative hypotheses for a chi-squared test with this example.
d. What are the degrees of
freedom?
e. The value of the chi square
test statistic is 7.61. Find the P-value. What is your conclusion?
Multiple choice (18 points)
16.
The appraised values of three recently
sold houses in the
A) 190.00 B)
27.84 C) 22.73
D) 16.07
17.
Does the mean
cost of text books per semester differ for math students and students in the
liberal arts? A sample of six math students and a sample of six liberal arts
students were asked how much their text books had cost the previous
semester.
|
A) |
the
matched-pairs t test. |
C) |
the two-sample t test |
|
B) |
the one-sample t test |
D) |
Any of the above are valid. |
18. Which of the
following statements is true?
|
A) |
Two-sample t procedures are less robust than the one-sample t
methods. |
|
B) |
In planning a two-sample study, it is best to choose equal sample
sizes. |
|
C) |
In planning a two-sample study, if the two population distributions
have different shapes, then you can use samples of size 5. |
|
D) |
None of the above is true. |
19. A teacher was interested in whether a
test-taking-skills class improved the pass rate on a high school exit exam. Let
be the proportion of
all students who took the skills class that passed the exit exam, and 
be the proportion of
students who did not take the skills class that passed the exit exam. A 95%
confidence interval for 
was calculated to be 
.
a.
The sampling distribution of the difference in
sample proportions 
has standard error equal to
|
A) |
0.055. |
|
B) |
0.1078. |
|
C) |
0.066. |
|
D) |
The standard error cannot be calculated without knowing the sample results. |
b.
Which of the following statements gives a
correct interpretation of the confidence interval?
|
A) |
We can be 95% confident that the difference between the sample proportions falls between –0.0578 and 0.1578. |
|
B) |
There is a 95% probability that the difference between proportions falls between –0.0578 and 0.1578. |
|
C) |
Ninety-five percent of the confidence intervals constructed will fall between –0.0578 and 0.1578. |
|
D) |
We can be 95% confident that the true difference in proportions falls between –0.0578 and 0.1578. |
c.
The teacher decides to use the confidence
interval to test the hypothesis 
using a 0.05 level of
significance. His decision should be:
|
A) |
reject the null hypothesis because 0 falls in the 95% confidence interval. |
|
B) |
fail to reject the null hypothesis because 0 falls in the 95% confidence interval. |
|
C) |
reject the null hypothesis because 0 does not fall in the 95% confidence interval. |
|
D) |
It is not possible to test the hypotheses using the 95% confidence interval. |
I have adhered to the
principles of the Honor Code in completing this test. __________________________