1 Math 140 Exam 2 COC Spring 2022 150 Points Question 1 (30 points) Match the following vocabulary words in the table below with the corresponding definitions. Confidence Interval Hypothesis Test Standard Error Alternative Hypothesis Randomized Simulation Random Sample Random Assignment Random Chance Population Sampling Variability Significance Level Type II Error One-Population Mean T-Test Statistic Quantitative Data One-Population Proportion Z-Test Statistic Categorical Data Critical Value Statistic Parameter Census Type I Error Bootstrap Distribution Margin of Error Beta Level Bootstrapping Null Hypothesis P-value Point Estimate a. A number we compare our test statistic to in order to determine significance. In a sampling distribution or a theoretical distribution approximating the sampling distribution, the critical value shows us where the tail or tails are. The test statistic must fall in the tail to be significant. b. Also called the Alpha Level. If the P-value is lower than this number, then the sample data significantly disagrees with the null hypothesis and is unlikely to have happened by random chance. This is also the probability of making a type 1 error. c. A statement about the population that does not involve equality. It is often a statement about a “significant difference”, “significant change”, “relationship” or “effect”. d. The collection of all people or objects you want to study. e. A number calculated from sample data in order to understand the characteristics of the data. f. When biased sample data leads you to support the alternative hypothesis when the alternative hypothesis is actually wrong in the population. g. Another word for sampling variability. The principle that random samples from the same population will usually be different and give very different statistics. h. Data in the form of numbers that measure or count something. They usually have units and taking an average makes sense. i. Taking many random samples values from one original real random sample with replacement. j. Collecting data from everyone in a population. 2 k. Collecting data from a population in such a way that every person in the population has an approximately equal chance of being chosen. This technique tends to give us data with less sampling bias. l. The probability of getting the sample data or more extreme because of sampling variability (by random chance) if the null hypothesis is true. m. The sample proportion is this many standard errors above or below the population proportion in the null hypothesis. n. Take a group of people or objects and randomly put them into two or more groups. This is a technique used in experiments to create similar groups. Similar groups help to control confounding variables so that the scientist can prove cause and effect. o. Data in the form of labels that tell us something about the people ...

1. 1
Math 140 Exam 2
COC Spring 2022
150 Points
Question 1 (30 points)
Match the following vocabulary words in the table below with
the corresponding definitions.
Confidence Interval Hypothesis Test Standard Error Alternative
Hypothesis
Randomized Simulation Random Sample Random Assignment
Random Chance
Population Sampling Variability Significance Level Type II
Error
One-Population Mean
T-Test Statistic
Quantitative Data One-Population
Proportion Z-Test
Statistic
Categorical Data

2. Critical Value Statistic Parameter Census
Type I Error Bootstrap Distribution Margin of Error Beta Level
Bootstrapping Null Hypothesis P-value Point Estimate
a. A number we compare our test statistic to in order to
determine significance. In a sampling
distribution or a theoretical distribution approximating the
sampling distribution, the critical
value shows us where the tail or tails are. The test statistic must
fall in the tail to be significant.
b. Also called the Alpha Level. If the P-value is lower than this
number, then the sample data
significantly disagrees with the null hypothesis and is unlikely
to have happened by random
chance. This is also the probability of making a type 1 error.
c. A statement about the population that does not involve
equality. It is often a statement about a
“significant difference”, “significant change”, “relationship” or
“effect”.
d. The collection of all people or objects you want to study.
e. A number calculated from sample data in order to understand
the characteristics of the data.

3. f. When biased sample data leads you to support the alternative
hypothesis when the alternative
hypothesis is actually wrong in the population.
g. Another word for sampling variability. The principl e that
random samples from the same
population will usually be different and give very different
statistics.
h. Data in the form of numbers that measure or count
something. They usually have units and
taking an average makes sense.
i. Taking many random samples values from one original real
random sample with replacement.
j. Collecting data from everyone in a population.
2
k. Collecting data from a population in such a way that every
person in the population has an
approximately equal chance of being chosen. This technique
tends to give us data with less
sampling bias.
l. The probability of getting the sample data or more extreme

4. because of sampling variability (by
random chance) if the null hypothesis is true.
m. The sample proportion is this many standard errors above or
below the population proportion in
the null hypothesis.
n. Take a group of people or objects and randomly put them into
two or more groups. This is a
technique used in experiments to create similar groups. Similar
groups help to control
confounding variables so that the scientist can prove cause and
effect.
o. Data in the form of labels that tell us something about the
people or objects in the data set.
p. The standard deviation of a sampling distribution. The
distance that typical sample statistics are
from the center of the sampling distribution. Since the center of
the sampling distributions is
usually close to the population parameter, the standard error
tells us how far typical sample
statistics are from the population parameter.
q. When someone takes a sample statistic and then claims that it
is the population parameter.
r. Two numbers that we think a population parameter is in

5. between. Can be calculated by either a
bootstrap distribution or by adding and subtracting the sample
statistic and the margin of error.
s. When biased sample data leads you fail to reject the null
hypothesis when the null hypothesis is
actually wrong in the population.
t. The sample mean is this many standard errors above or below
the population mean in the null
hypothesis.
u. Putting many bootstrap statistics on the same graph in order
to simulate the sampling variability
in a population, calculate standard error, and create a
confidence interval. The center of the
bootstrap distribution is the original real sample statistic.
v. The probability of making a type 2 error.
w. A statement about the population that involves equality. It is
often a statement about “no
change”, “no relationship” or “no effect”.
x. Random samples values and sample statistics are usually
different from each other and usually
different from the population parameter.
y. Total distance that a sample statistic might be from the

6. population parameter. For normal
sampling distributions and a 95% confidence interval, the
margin of error is approximately twice
as large as the standard error.
z. A number that describes the characteristics of a population
like a population mean or a
population percentage. Can be calculated from an unbiased
census, but is often just a guess
about the population.
aa. A procedure for testing a claim about a population.
bb. A technique for visualizing sampling variability in a
hypothesis test. The computer assumes the
null hypothesis is true, and then generates random samples. If
the sample data or test statistic
falls in the tail, then the sample data significantly disagrees
with the null hypothesis. This
technique can also calculate the P-value without a formula.
3

7. Question 2 (25 Points)
2-Population Mean Hypothesis Test
a. Determine if the following two-population mean tests are
matched pair or independent groups.
b. Write the null and alternative hypothesis. Include the claim
and what type of test.
c. Check all of the assumptions for a two-population mean T-
test. Explain your answers. Does the
problem meets all the assumptions?
d. Write a sentence to explain the T-test statistic.
e. Use the test statistic and the critical value to determine if the
sample data significantly disagrees
with the null hypothesis. Explain your answer
f. Write a sentence to explain the P-value.
g. Use the P-value and significance level to determine if the
sample data could have occurred by
random chance (sampling variability) or is it unlikely to random
chance? Explain your answer.
h. Should we reject the null hypothesis or fail to reject the null
hypothesis? Explain your answer.
i. Write a conclusion for the hypothesis test. Explain your
conclusion in plain language.
j. Is the categorical variable related to the quantitative variable?

8. Explain your answer.
Here is the scenario.
Researchers collected data on traffic flow, number of shoppers,
and traffic accident-related emergency
room admissions on Friday the 13th and the previous Friday,
Friday the 6th. They randomly selected 40
Friday the 13ths and 40 Friday the 6ths over a ten-year period.
Also given are some sample statistics,
where the difference is the number of cars on the 6th minus the
number of cars on the 13th. We claim
that the difference is not zero. (Use a 5% significance level).
Friday 6th Friday 13th Difference
�̅� 128,385 126,550 1835
� 7,664 7,259 405
� 40 40 40
The T-test statistic is ±4.94 and critical value is 2.38. The p-
value is < 0.01.
Question 3 (15 Points)
2-Population Mean Confidence Interval
a. Does the data meet the assumptions for inference with two
population proportions or two

9. population means? If it is two means, are the groups
independent or matched pair? List the
assumptions needed and how the problem meets them or does
not meet them.
b. Does the confidence interval indicate that the mean from
population 1 is higher, lower, or not
significantly different from population 2? Explain how you
know.
c. Write the two-population confidence interval sentence
explaining this confidence interval.
The scenario is the same as in question 2. Also included here is
the confidence interval (1518, 2151)
with confidence level 95%.
4
Question 4 (20 Points)
1-Population Proportion Hypothesis Test
a. Give the null and alternative hypothesis. Which is the claim?
Is this a right-tailed, left-tailed, or
two-tailed test? Explain how you know what tail to use.
b. Check the assumptions.

10. c. Give the test statistic and write the standard sentence to
explain it. Compare your test statistic
to the critical value. Did the sample data significantly disagree
with the null hypothesis? Explain
how.
d. Give the p-value and write the definition sentence to explain
it. Could the sample data have
happened because of sampling variability (random chance) or is
it unlikely to be sampling
variability? Explain why.
e. Compare the p-value to the significance level. State whether
you reject the null hypothesis or
fail to reject the null hypothesis. Explain your answer.
f. Write the standard conclusion.
g. Explain your conclusion in easy to understand language.
Here is the scenario.
An online source suggests that one out of every three people in
the U.S have high blood pressure and
the population proportion of U.S. adults is 33.3%. Another
website disagrees with this and claims that
the true percentage of U.S. adults with high blood pressure is
dramatically lower than 1 in 3 (33.3%). A

11. random sample of 500 U.S. adults found that 165 of them had
high blood pressure. Use the printout and
a 10% significance level to test the claim that less than 33% of
U.S. adults have high blood pressure.
N Sample
Proportion
Significance
Level
Critical Value Test Statistic Z P-value
500 0.33 0.10 -1.282 -0.142 0.4434
Question 5 (10 Points)
1-Population Proportion Bootstrap Confidence Interval
a. Does the data meet the assumptions for a bootstrap
confidence interval? Explain your answer.
b. How many bootstrap samples were taken (see picture below)?
c. What is the shape of the bootstrap distribution?
d. Write the upper and lower limits of the bootstrap confidence
interval.
e. Write a sentence to explain the bootstrap confidence interval
estimate of the population
proportion.

12. Here is the scenario.
An experiment employing randomization was conducted to see
what percentage of rats would show
empathy toward fellow rats in distress. Of the 30 total rats in
the study, 23 showed empathy. Use the
bootstrap distribution to find a 99% confidence interval for the
population proportion.
5
Question 6 (20 Points)
a. State the Central Limit Theorem for Means and explain the
ideas behind it.
b. Describe the process of making a sampling distribution.
c. What is a point estimate? Discuss how point estimates create
confusion for people reading
articles and scientific reports.
d. What conditions should be met to ensure that a sampling
distribution of sample proportions is
normal?

13. Question 7 (5 Points)
State the assumptions for a one-population variance or standard
deviation confidence interval.
6
Question 8 (5 Points)
Use the following formulas to identify the sample statistic (p-
hat or x-bar or s) and the margin of error.
������ ��������� =
(����� ����� + ����� �����)
2
������ �� ����� =
(����� ����� − ����� �����)
2
A 95% confidence interval estimate of the population proportion
of pies in a county fair in Ohio is
(0.46, 0.52).
Question 9 (20 Points)
Type I and II Errors

14. Scenario I:
Mike and his advertisement team have created an advertisement
plan for a new flavor of soda. Right
now, approximately 16% of soda drinkers are purchasing this
flavor. Mike needs to show his bosses that
his advertisement plan will increase the percentage of soda
drinkers purchasing this new flavor. If Mike’s
advertising team succeeds in increasing the percentage of
customers that prefer this new flavor, then
the company will increase supply and make more of the soda to
meet demand. If not, then the company
will keep the supply as it currently is. After the advertising
changes, Mike takes a random sample of
customers to determine if the percentage of soda drinkers that
like the new flavor has increased. (They
are currently using a 5% significance level).
�0: � = 0.16 (�ℎ � ������� ���� ��� ��������
���������� �� �ℎ � ��� ������ �� ����. )
��: � > 0.16 (�ℎ � ������� ���� ��������
���������� �� �ℎ � ��� ������ �� ����. )
a. Write a description of a type 1 error and the possible
consequences of that error in the context
of the problem.

15. b. Write a description of a type 2 error and possible
consequences of that error in the context of
the problem.
c. Would you recommend any changes to the significance level
or sample size based on what you
know about the type 1 and type 2 errors in this problem?
Explain.
Scenario II:
A global sportswear company is contemplating contributing
money to a political candidate in the next
election. The managers of the company do not want to
contribute unless they are sure the candidate
will get the majority of the population vote and win the
election. Otherwise, the company will not
contribute to the candidates’ campaign. (They are currently
using a 10% significance level).
�0: � ≤ 0.5 (�ℎ � ������� ���� ���
���������� �� �ℎ � ��������� ����������
′ ��������. )
��: � > 0.5 (�ℎ � ������� ���� ���������� ��
�ℎ � ��������� ����������
′ ��������. )
7

16. a. Write a description of a type 1 error and the possible
consequences of that error in the context
of the problem.
b. Write a description of a type 2 error and possible
consequences of that error in the context of
the problem.
c. Would you recommend any changes to the significance level
or sample size based on what you
know about the type 1 and type 2 errors in this problem?
Explain.