The document provides information and questions related to statistics concepts including hypothesis testing, probability, sampling, and data analysis. It includes sample size, means, standard deviations, test statistics, critical values, and confidence levels for hypothesis tests. Questions involve probabilities and distributions for dice rolls, sports preferences among university students, apple weights, disease screening tests, fraud detection using Benford's law, and gemstone hardness tests. Other questions cover seating arrangements, swimming pool contamination sampling, and determining the number of dishes to order for a banquet.

1. STAT
Use the information below to answer Questions 1 through 4.
Given a sample size of 36, with sample mean 670.3 and sample
standard deviation 114.9, we perform the following hypothesis
test.
Null Hypothesis
Alternative Hypothesis
What is the test statistic?
At a 10% significance level (90% confidence level), what is the
critical value in this test? Do we reject the null hypothesis?
What are the border values between acceptance and rejection of
this hypothesis?
What is the power of this test if the assumed true mean were
710 instead of 700?
Questions 5 through 8 involve rolling of dice.
Given a fair, six-sided die, what is the probability of rolling the
die twice and getting a “1” each time?
What is the probability of getting a “1” on the second roll when
you get a “1” on the first roll?
The House managed to load the die in such a way that the faces
“2” and “4” show up twice as frequently as all other faces.
Meanwhile, all the other faces still show up with equal
frequency. What is the probability of getting a “1” when rolling
this loaded die?

2. Write the probability distribution for this loaded die, showing
each outcome and its probability. Also plot a histogram to show
the probability distribution.
Use the data in the table to answer Questions 9 through 11.
x
3
1
4
4
5
y
1
-2
3
5
9
Determine SS
xx
, SS
xy
, and SS
yy
.
Find the equation of the regression line. What is the predicted
value when
Is the correlation significant at 1% significance level (99%

3. confidence level)? Why or why not?
Use the data below to answer Questions 12 through 14.
A group of students from three universities were asked to pick
their favorite college sport to attend of their choice: The results,
in number of students, are listed as follows:
Football
Basketball
Soccer
Maryland
60
70
20
Duke
10
75
15
UCLA
35
65
25
Supposed a student is randomly selected from the group
mentioned above.
What is the probability that the student is from UCLA or
chooses football?
What is the probability that the student is from Duke, given that

4. the student chooses basketball?
What is the probability that the student is from Maryland and
chooses soccer?
Use the information below to answer Questions 15 and 17.
There are 3600 apples in a shipment. The weight of the apples
in this shipment is normally distributed. It is found that it a
mean weight of 14 ounces with a standard deviation of 2.5
ounces.
How many of apples have weights between 13 ounces and 15
ounces?
What is the probability that a randomly selected mango weighs
less than 12.5 ounces?
A quality inspector randomly selected 100 apples from the
shipment.
What is the probability that the 100 randomly selected apples
have a mean weight less than 12.5 ounces?
Do you come up with the same result in Question 16? Why or
why not?
A pharmaceutical company has developed a screening test for a
rare disease that afflicted 2% of the population. Unfortunately,
the reliability of this test is only 80%, which means that 20% of
the tested will get a false positive. If a subject is tested positive
based on this test, what is the probability that he has the
disease?
Use the information below to answer Questions 19 and 20.
Benford's law
, also called the
first-digit law
, states that in lists of numbers from many (but not all) real-life

5. sources of data, the leading digit is distributed in a specific,
non-uniform way shown in the following table.
Leading Digit
1
2
3
4
5
6
7
8
9
Distribution of Leading Digit (%)
30.1
17.6
12.5
9.7
7.9
6.7
5.8
5.1
4.6
The owner of a small business would like to audit its account

6. payable over the past year because of a suspicion of fraudulent
activities. He suspects that one of his managers is issuing
checks to non-existing vendors in order to pocket the money.
There have been 790 checks written out to vendors by this
manager. The leading digits of these checks are listed as follow:
Leading Digits
50
15
12
74
426
170
11
23
9
Suppose you are hired as a forensic accountant by the owner of
this small business, what statistical test would you employ to
determine if there is fraud committed in the issuing of checks?
What is the test statistic in this case?
What is the critical value for this test at the 5% significance
level (95% confidence level)? Do the data provide sufficient
evidence to conclude that there is fraud committed?

7. Two different simple random samples are drawn from two
different populations. The first sample consists of 20 people
with 10 having a common attribute. The second sample consists
of 2000 people with 1404 of them having the same common
attribute.
Perform a hypothesis test of
p
1
=
p
2
with a 5% significance level (95% confidence level).
Obtain a 95% confidence interval estimate of
p
1
-
p
2
. Do you come up with the same conclusion for Question 21?
Why or why not?
95% confidence interval estimate of
p
1
-
p
2
is,
Hardness of Gem – Questions 23 and 24
Listed

8. below are measured hardness indices from three different
collections of gemstones.
Collection
Hardness Indices
A
9.3
9.3
9.3
8.6
8.7
9.3
9.3
--
---
---

9. ---
---
---
9.91
0.10
B
8.7
7.7
7.7
8.7
8.2
9.0
7.4
7.0
---
---
---
---
---
8.03
0.60
C
7.2
7.9
6.8
7.4
6.5
6.6
6.7
6.5
6.5
7.1
6.7
5.5
7.3
6.82

10. 0.34
You are also given that
.
What is the test statistic?
Use a 5% significance level (95% confidence level) to test the
claim that the different collections have the same mean
hardness.
A couple has 3 daughters. The wife is expecting another baby.
What is the probability that the new baby is a girl again?
Suppose the new baby turns out to be a girl. What is the
probability that a family with 4 children that are all girls?
Use the data below to answer Questions 26 and 27.
This is a summary of the midterm scores for two sections of
STAT 230. The midterm questions and the grading criteria are
different in these two sections.
Section A
Student
Score

11. Section B
Student
Score
A
70
H
15
B
42
I
57
C
53
J
48
D
61
K
90
E
22
L
85
F
85
M
73
G
59
N
49
-----

12. ------
O
39
What are the mean and standard deviation of the scores in
Section A?
We notice that Student
F
in Section A and Student
L
in Section B have the same numerical score.
How do they stand relative to their own classes?
Which student performed better? Explain your answer.
Composite sampling is a way to reduce laboratory testing costs.
A public health department is testing for possible fecal
contamination in public swimming pools. In this case, water
samples from 5 public swimming pools are combined for one
single test, and further testing is performed only if the
combined sample shows fecal contamination. Based on past
experience, there is a 3% chance of finding fecal contamination
in a public swimming area. What is the probability that a
combined sample from 5 public swimming pools has fecal
contamination?
Peter, Paul, Mary, Andrew, John, and Martha are members of
the pastoral council at a local church. They are to be seated at
one side of a long conference table in a pastoral council
meeting.
How many possible ways can these 6 council members can be

13. seated?
How many possible sitting arrangements are there if only
gender is considered in the process?
A banquet organizer knows that not all 600 invited guests will
show up at an event. Based on past experience, only 80% of the
invited guest for this special event will come. When expensive
dishes are served, it would be prudent not to order the full 600
plates because a good number of them will be wasted. On the
other hand, the banquet organizer will try to stay within 7%
probability that he would not have to rush to prepare the
expensive dishes. How many of these expensive dishes would
you order if you were organizing this banquet?