1.
2.
If you’ve ever played DD (Dungeons and Dragons) then you’ve seen a four-sided die. The probability of
getting any of the results (1, 2, 3, or 4) is 0.25. Suppose you tossed a four-sided die 10,000 times.
a.
Let p̂ be the proportion of the 10,000 tosses which result in a 3. What is the mean and standard
deviation of the distribution of p̂ ?
b.
What is the probability that you would get fewer than 2450 3’s?
In 2007, the average claim amount requested at GEICO Insurance is for $1400 with a standard deviation of
$300. Furthermore, all claims are distributed in an approximate normal shape. Suppose you took a random
sample of 100 GEICO Insurance claims and averaged them.
a.
Let x be the mean of the 100 claims. What is the mean and standard deviation of the distribution
of all such x ’s?
b.
What is the probability that the sample mean is greater than $1450?
c.
Can we state that there is a 50% chance that any one claim drawn randomly would have a value
that was less than or equal to $1400? Explain.
3.
4.
The City of Burlington is interested in how many people in the city are in favor of consolidating several of
the elementary schools. Effectively this would cause certain schools to be closed down and students from
these neighborhoods would need to be bussed into other neighborhood schools which would remain open.
The city randomly contacts 400 individuals who live in the city and polls them on this issue. Of the four
hundred, 120 are opposed to the issue and 280 are in favor.
a.
Construct a 95% confidence interval for the percentage of the City of Burlington residents who are
in favor of school consolidation.
b.
A supporter of schools not being consolidated contests your data. He states that he visited 400
people in a specific neighborhood around one of the schools that is proposed as being shut down.
He found that 350 of those visited opposed the consolidation while only 50 were in favor. He
argues that the City of Burlington should use his data instead of their data. Why should the City be
skeptical? Explain.
A sociologist is interested in how many hours a week families spend talking with each other. She randomly
polls 6400 families and finds that the average number hours that family’s talk during a given week (7-day
period) is 18 hours with a standard deviation of 12.2 hours. Construct a 99% confidence interval for the
mean number of hours in which families talk with each other during a week.
5.
The average height of an 18 - 21 year old male in the U.S. is 68.4 inches, as stated by Wikipedia. We will
test whether or not the average height of an undergraduate male student at UVM differs significantly from
this. 144 male students are randomly selected and it is found that the average height is 68.1 inches with a
standard deviation of 5.2 inches.
a.
State the null and alternative hypotheses, where μ is the average height of a UVM student.
b.
Assuming the null hypothesis is true, compute the z-value for our data.
c.
Find the p-value for this problem.
d.
Use an alpha level of 0.05 to determine if our data significant enough to reject our null hypothesis?
e.
Use the data from the 144 students randomly selected to find a 95% confidence interval for the
average height of a male UVM undergraduate student. Give the interval in the form x  E where
E is the margin of error. Round your margin of error to three decimal places.
6.
Treehugger.com states that about 77% of all people in the U.S. recycle on a regular basis. We will test
whether or not the proportion of undergraduate students at UVM who recycle is more than this. 400
students are randomly chosen and 360 say that they recycle on a regular basis.
a.
State the null and alternative hypotheses, where p is the proportion of UVM students who recycle.
b.
Assuming the null hypothesis is true, compute the z-value for our data.
c.
Find the p-value for this problem.
d.
Use an alpha level of 1% to determine if our data significant enough to reject our null hypothesis?
e.
Use the data from the 400 students randomly polled to find a 99% confidence interval for the
proportion of UVM undergraduate students who recycle. Give the interval in the form p̂  E
where E is the margin of error. Round your margin of error to three decimal places.
7.
A study examined what children in grades 4, 5, and 6 view as being more important, sports or academics. A
random sample of students from a Georgia school district was selected from these grade levels. The
students received a questionnaire regarding preference. They were asked what they felt was more
important in life: to be a good athlete (Athletics) or to be a good student (Academics). Results are
presented in the following table broken down by the stated gender of the child.
Academics
Athletics
Boys
285
56
Girls
295
45
a.
What percentage of the boys stated that academics were more important?
b.
What percentage of the girls stated that academics were more important?
c.
Fill in the blank in the following sentence.
Based on data collected, elementary school girls in grades 4, 5, and 6 are __________ times as
likely to think that academics are more important that elementary school boys.
d.
You plan to do a Chi-Square Test of Independence to determine if there is a significant difference in the
perception of academics versus athletics when boys are compared to girls. The null hypothesis for this
situation is, “Gender is independent of preference of academics and athletics.” What is the alternative
hypothesis?
e.
Assuming the null hypothesis is true, find the expected value for each cell in the table.
f.
Find the chi-square statistic for this data.
g.
Using the table given on-line or the one found in the book; give the interval in which the p-value lives.
h.
Is the data significant enough to claim that there is a difference in the perception of academics versus
athletics when boys are compared to girls? Use a significance level of 5%.
8.
9.
Studies show that it is known that 24% of all students at a university/college smoke. Suppose that we focus
on a random sample of 400 students at a particular university/college.
a.
Let p̂ be the proportion of the sample of size 400 who smoke. What would be the mean value of
the distribution of all possible p̂ ’s?
b.
Let p̂ be the proportion of the sample of size 400 who smoke. What would be the standard
deviation of the distribution of all possible p̂ ’s?
c.
Is the distribution of all possible p̂ ’s guaranteed to be normal? Explain.
d.
What is the probability that at least 26% of the students in the dorm smoke?
e.
What is the probability that no more than 20% of the students in the dorm smoke?
Chemists use pH to measure the acidity/alkaline nature of compounds. A 50-gallon drum of mixed
commercial chemicals is supposed to have a mean pH of   6.3 with a standard deviation of   1.9 .
Suppose that you randomly draw 36 50-gallon drums from the production line and compute their pH. Let
x be the average pH of the 36 drums.
a.
What would be the mean value of the distribution of all possible x ’s? What would be the standard
deviation of the distribution of all possible x ’s?
b.
Is the distribution of all possible x ’s guaranteed to be normal? Explain.
c.
What is the probability that the sample mean pH will be less than 6?
d.
What is the probability that the sample mean pH will be greater than 6.8?
e.
Suppose that the sample mean came out to be x  7.4 . What should you most likely conclude?
10. A random sample of 100 evenings (6 PM to 9 PM) at the O’Sullivan household showed the family received
an average of x  5.2 phone calls each evening. The sample standard deviation was s  1.2 . Find a 95%
confidence interval for the actual mean number of phone calls received each evening.
11. A large furniture store has begun a new ad campaign on local television. Before the campaign, the term
average daily sales were $24,819. The store is interested in whether the average daily sales is now greater
than $24,819. A random sample of 40 days during the new ad campaign gave a sample mean daily sale of
x  $25,910 with a sample standard deviation of s  $1917 .
a.
State the null and alternative hypotheses.
b.
Find the z-value for our data assuming the null hypothesis is true.
c.
Make a conclusion regarding the hypotheses stating the p-value. Use a significance level of 5%.
12. Recently the national average yield on municipal bonds has been   4.19% . A random sample of 100
Arizona municipal bonds gave an average yield of 5.11% with sample standard deviation of s  1.15% .
Does this indicate that the mean yield for all Arizona municipal bonds is greater than the national average
of 4.19%?
a.
State the null and alternative hypotheses.
b.
Find the z-value for our data assuming the null hypothesis is true.
c.
Make a conclusion regarding the hypotheses stating the p-value. Use a significance level of 5%.
13. At a local four-year college, 37% of the student-body is freshmen. A random sample of 42 student names
taken from the Dean’s Honor List over the past several semesters showed that 17 were freshmen. Does
this indicate the population proportion of freshmen on the Dean’s Honor List is significantly different from
37%?
a.
State the null and alternative hypotheses.
b.
Find the z-value for our data assuming the null hypothesis is true.
c.
Make a conclusion regarding the hypotheses. Use a significance level of 1%.
14. A study was done to see if men or women are more likely to come to someone’s aid. A researcher entered
an elevator and dropped a handful of pencils, with the appearance that it was an accident. The question
was whether the males or females who observed the mishap would be more likely to help pick up the
pencils. The results are shown in the table below.
Count
Male Observer
Female Observer
Total
Helped Pick Up Pencils
Yes
No
Total
370
950
1320
300
1003
1303
670
1953
2623
a.
What percentage of males helped pick up pencils? Round to the nearest percentage.
b.
What percentage of females helped pick up pencils? Round to the nearest percentage.
c.
What is the relative risk of picking up pencils when males are compared to females? Round your answer to
two decimal places and use it in a sentence.
d.
Name one unknown variable which could cause some bias in this study.
e.
Compute the expected count for each cell in the table. Round your answers to two decimal places and
place your answers on the given lines.
f.
Find the chi-square statistic for this set of data. Round your answer to two decimal places.
g.
Make a conclusion about the data based on the chi-square statistic. Use a significance level of 5%.