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College of the Canyons
Exam 3 Fall 2010, V3
Amy Morrow
Name:
Answer the following questions NEATLY. Show all necessary work directly on the exam.
Scratch paper will be discarded unread. One point each part unless otherwise marked.
1. A certain population is approximately normal. We want to estimate its mean, so we will collect a sample.
Which should be true if we use a large sample rather than a small one?
I. The distribution of our sample data will be approximately normal.
II. The sampling distribution of the sample means will be approximately normal.
III. The variability of the sample means will be smaller.
(Circle the best answer)
A. II only
B. III only
C. II and III
D. I only
E. I, II, and III
2. A can of pumpkin pie mix contains a mean of 30 ounces and a standard deviation of 2 ounces. The
contents of the cans are normally distributed. What is the probability that four randomly selected cans
of pumpkin pie mix contain a total of more than 126 ounces? Include an appropriately labeled and
shaded diagram. (2 points)
Solution:
Note: This is equivalent to the cans having an
average amount of more than 31.5 ounces. Using the central limit theorem, we use the normal
model with mean 30, SD = √24 = 1, we are looking for the probability that the mean amount of
cans is more than 31.5. ANSWER: 0.067.
3. To plan the course offerings for the next year, a university department dean needs to estimate what
impact the “No Child Left Behind” legislation might have on the teacher credentialing program. Historically, 40% of this university’s pre-service teachers have qualified for paid internship positions each
year. The Dean of Education looks at a random sample of internship applications to see what proportion
indicate the applicant has achieved the content-mastery that is required for the internship. Based on
these data, he creates a 90% confidence interval of (33%, 41%). Could this confidence interval be used
to test the hypothesis H0 : p = 0.40 versus HA : p < 0.40 at the α = 0.05 level of significance? (Circle
the best answer)
A. Yes, since 40% is not in the center of the confidence interval, he rejects the null hypothesis,
concluding that the percentage of qualified applicants will change.
B. Yes, since 40% is in the confidence interval he fails to reject the null hypothesis,
concluding that there is not strong enough evidence of any change in the percent
of qualified applicants.
C. No, because the dean only reviewed a sample of the applicants instead of all of them.
D. No, the salaries of paid internships might not be normally distributed.
College of the Canyons
Exam 3 Fall 2010, V3
Amy Morrow
4. On many highways state police officers conduct inspections of driving logbooks from large trucks to see
if the trucker has driven too many hours in a day. At one truck inspection station they issued citations
to 49 of 348 truckers that they reviewed.
(a) Based on the results of this inspection station, construct a 95% confidence interval for the proportion
of truck drivers that have driven too many hours in a day.
Solution: (.104, .177) or (.106, .182)
(b) Interpret the interval you provided.
Solution: We are 95% confident that between 10% and 18% of truckers have driven too many
hours in a day.
(c) Is there evidence to suggest that more than 10% of truckers have driven too many hours in a day?
Explain.
Solution: Yes, the interval is above 10%.
5. Based on a random sample of 50 students, the 90 percent confidence interval for the mean amount of
money students spend on lunch at a certain high school is found to be ($3.45, $4.15). Which of the
following statements is true? (Circle the best answer)
A. 90% of all random samples of 50 students obtained at this high school would result in a sample
mean amount of money students spend on lunch between $3.45 and $4.15.
B. 90% of the time, the mean amount of money that all students spend on lunch at this high
school will be between $3.45 and $4.15.
C. 90% of all students spend between $3.45 and $4.15 on lunch at this high school.
D. About 90% of all random samples of 50 students obtained at this high school
would result in a confidence interval that contains the true mean amount of money
students spend on lunch.
E. Approximately 45 of the 50 students in the random sample will spend between $3.45 and $4.15
on lunch at this high school.
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College of the Canyons
Exam 3 Fall 2010, V3
Amy Morrow
6. A survey conducted of 100 randomly selected people in the United States yielded the results shown in
the table below for favorite color. Is there evidence that the distribution of favorite color is different for
each gender?
Blue Red Yellow
Male
25
20
25
Female
5
10
15
(a) Name the test.
Solution: test of homogeneity
(b) Hypotheses. (2 points)
Solution: H0 : The distribution of favorite color is the same for each gender.
HA : They are different
(c) Check conditions.
Solution: Independence - assumed. Random - stated in problem. Expected cells - smallest is
9, which is more than 5. Counted data - check.
(d) Draw, label, and shade model. (2 points)
Solution:
This
is
a
χ2
distribution
with
2
df.
(e) Test statistic and p-value. (2 points)
Solution: χ2 = 3.770, p = 0.152
(f) State conclusions (both statistical and experimental). (2 points)
Solution: Because the p-value is large, we fail to reject H0 . There is no evidence that the
distributions are different.
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College of the Canyons
Exam 3 Fall 2010, V3
Amy Morrow
7. A father is concerned that his teenage son is watching too much television each day since his son watches
an average of 2 hours per day. His son says this his TV habits are no different than those of his friends.
Since this father has taken an intro stats class, he knows that he can actually test to see whether or not
his son is watching more TV than his peers. The father collects a random sample of television watching
times from boys at this son’s high school and gets the following data:
1.9 2.3 2.2 1.9 1.6 2.6 1.4 2.0 2.0 2.2
Is the father right? That is, is there evidence that the other boys average less than 2 hours of television
per day? Conduct a hypothesis test.
(a) Hypotheses. (2 points)
Solution: H0 : µ = 2, HA : µ < 2
(b) Check conditions.
Solution: Independent, randomization, 10% - check. Nearly normal - n < 10, so we must
check the histogram and normal probability plot. The histogram is unimodal and reasonably
symmetric. The normal probability plot looks good, so CHECK.
(c) Draw, label, and shade model. (2 points)
Solution:
Technically, this would be a t-model with 9 df,
but I will draw the normal because minitab
√
does that better. Mean = 2, SE = 0.345
=
10
0.109.
NOTE: as we can see, the data is on the wrong
side, so we’re expecting to fail to reject H0 .
(d) Test statistic and p-value. (2 points)
Solution: t = 0.092. pval = 0.536
(e) State conclusions (both statistical and experimental). (2 points)
Solution: Fail to reject H0 . There is no evidence that the other boys watch less than 2 hours,
the amount watched by the son.
(f) As the son who is worried about his television time being restricted, which kind of error is worse
to you?
Solution: Type I. If we find other boys watch less, the father might restrict tv time.
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College of the Canyons
Exam 3 Fall 2010, V3
Amy Morrow
8. A pharmaceutical company investigating whether food markets or drug stores are less likely to remove
over-the-counter drugs from the shelves when the drugs are past the expiration date found a P-value of
2.8%. This means that: (Circle the best answer)
A. Food markets remove 2.8% more over-the-counter drugs from the shelves when the drugs are
past the expiration date than drug stores.
B. None of these.
C. There is a 2.8% chance that the food markets remove more expired over-the-counter drugs.
D. 2.8% of food markets remove over-the-counter drugs from the shelves when the drugs are past
the expiration date
E. There is a 2.8% chance the drug stores remove more expired over-the-counter drugs.
9. A farmer would like to know if a new fertilizer increases his crop yield. In an effort to decide this, the
farmer recorded the yield for 10 different fields prior to adding fertilizer and after adding fertilizer. What
procedure will the farmer need to use to analyze his data? (Circle the best answer)
A. A procedure for analyzing the difference of proportions, two samples.
B. A procedure for analyzing the difference of two means, independent samples.
C. A procedure for analyzing the difference of two means, matched pair samples.
D. A procedure for analyzing the mean, one sample.
E. A procedure for analyzing the proportion, one sample.
F. A procedure for analyzing the difference of proportions, two samples.
10. A professor at a large university believes that students take an average of 15 units per term. A random
sample of 24 students in her class of 250 students reported the following number of credit hours that they
were taking. Does this sample indicate that students are taking more credit hours than the professor
believes?
9
9 12 12 13 13 14 14 14 15 15 15
15 15 15 16 16 16 16 17 17 17 18 18
(a) Hypotheses (2 points)
Solution: H0 : µ = 15, HA : µ > 15
(b) Does this sample indicate that students are taking more credit hours than the professor believes?
Provide statistical evidence to support your conclusion. (2 points)
Solution: The sample mean is 14.625, which is less than 15 (not more), so NO evidence. Note:
The p-value is 0.775, and the 95% CI is (13.615, 15.635).
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