Download ccc

Name: ___________________________________Score: ___________
MATH-138: Exam #3 – Version A
Note: Please write your answers clearly – unreadable answers will be marked incorrect.
You may use a calculator, StatCrunch, and/or one (2-sided) 4x6” index card. Please show
your work to receive full credit. The exam has 100 points total.
1. (2 points each) Identify the following statements as either true or false:
a. It’s possible to simultaneously make a Type I and a Type II error when performing a
hypothesis test.
False
b. Just like p-values, test statistics are always positive.
False
c. The p-value is the probability that H0 is true.
False – the p-value is the conditional probability that our observations occur given
that the null-hypothesis is true.
d. For a given sample size, a higher confidence level means a larger margin of error.
True
e. The standard deviation of the sampling distribution of Y increases as the sample size
increases.
False
2. A common HCC rumor/hypothesis is that the second MATH-138 unit exam is harder
than the first. To test this hypothesis, a random sample of 10 spring 2011 HCC MATH138 students was sampled and the results of their first two MATH-138 unit exams are
given in the table below. Does this sample provide evidence that the second unit exam is
indeed harder than the first? Let alpha=0.05 and assume all the conditions of the test are
met.
Student
Unit 1 Exam Score
Unit 2 Exam Score
1
85
84
2
79
77
3
92
93
4
77
79
5
97
76
6
87
53
7
85
63
8
88
88
9
83
87
10
89
74
a. (2 points) Type of test:
2-Mean Hypothesis Test (T-Test) (paired samples)
b. (4 points) Hypotheses:
H0: μd=0
HA: μd>0
c. (2 points) State the p-value and the value of the test statistic (2 decimal places each):
p-value: 0.03
t: 2.12
d. (1 point) Draw a sketch of the distribution of the test statistic. Label the value of the
test statistic from Part C above, and shade in the p-value area from Part C above:
e. (2 points) State whether or not the null hypothesis should be rejected:
Reject
f. (3 points) State a conclusion in the context of the problem: The statistical evidence
indicates that the second exam is very likely harder than the first.
3. The following contingency table shows IQ score results for a simple random sample of
500 Steelers and Ravens fans. Determine if intelligence and football team preference are
related (i.e. dependent). Let alpha=0.05 and assume all the conditions of the test are met.
Low IQ Score (<100)
High IQ Score (>100)
Total
Steelers Fans
116
234
350
Ravens Fans
82
68
150
Total
198
302
500
a. (2 points) Type of test:
χ2 Test of Independence
b. (4 points) Hypotheses:
H0: Intelligence & football team preference are independent
HA: Intelligence & football team preference are dependent
c. (2 points) State the p-value and the value of the test statistic (2 decimal places each):
p-value: 6.49E-6
χ2: 20.34
d. (1 point) Draw a sketch of the distribution of the test statistic. Label the value of the
test statistic from Part C above, and shade in the p-value area from Part C above:
e. (2 points) State whether or not the null hypothesis should be rejected:
Reject
f. (3 points) State a conclusion in the context of the problem: The statistical evidence
indicates that intelligence and football team preference appear to be related.
4. (REMOVED FROM EXAM BUT FYI)
The following data lists the frequency counts for the birth months of the 43 United States
Presidents:
Birth Month
# Presidents
Jan
4
Feb
4
Mar
4
Apr
4
May
2
Jun
1
Jul
4
Aug
5
Sep
1
Oct
6
Nov
5
Even though the “expected cell frequency condition” is technically violated with this
example, test the hypothesis (let alpha=0.10) that presidential birthdays are uniformly
spread across the months. Assume that all of the other conditions of the test are met.
Also, assume that a randomly sampled person has an equal chance (1 out of 12) of being
born in any given month1.
a. (2 points) Type of test:
χ2 Test of Goodness of Fit
b. (4 points) Hypotheses:
H0: Presidential birth months are spread out evenly throughout the year
HA: Presidential birth months are not spread out evenly throughout the year
c. (2 points) State the p-value and the value of the test statistic (2 decimal places each):
p-value: 0.76
χ2: 7.51
d. (2 points) Draw a sketch of the distribution of the test statistic. Label the value of the
test statistic from Part C above, and shade in the p-value area from Part C above:
e. (1 point) State whether or not the null hypothesis should be rejected:
Fail to Reject
f. (3 points) State a conclusion in the context of the problem: There is not enough
statistical evidence to suggest that presidential birth months are not spread out
evenly throughout the year.
1
This assumption is probably not true since some months have more days than others.
Dec
3
5. A “regional jet” is a small, jet aircraft that airlines increasingly use to fly their short
and medium-length routes. The number of passenger seats for 10 randomly sampled
regional jet types are shown below:
Regional Jet Type
# Passenger Seats
ARJ
107
AR7
70
CR1
100
CRJ
50
EM4
50
ERJ
50
ER4
50
FRJ
32
F70
78
141
75
Perform a hypothesis test (let alpha=0.05) to test the claim that the mean number of seats
of all regional jets is less than 75. Assume all the conditions of the test are met.
a. (2 points) Type of test:
1-Mean Hypothesis Test
b. (4 points) Hypotheses:
H0: μ=75
HA: μ<75
c. (2 points) State the p-value and the value of the test statistic (2 decimal places each):
p-value: 0.14
t: -1.15
d. (1 point) Draw a sketch of the distribution of the test statistic. Label the value of the
test statistic from Part C above, and shade in the p-value area from Part C above:
e. (2 points) State whether or not the null hypothesis should be rejected:
Fail to Reject
f. (3 points) State a conclusion in the context of the problem: There is not enough
statistical evidence to indicate that the true population mean number of passenger
seats in regional jet aircraft is less than 75.
6. Suppose you want to estimate (via a confidence interval) the population proportion of
Columbia, MD households that have a household income greater than $70,000.
a. (6 points) How many households would you have to sample to be 98% confident that
your confidence interval contains the true population proportion with a margin of error of
0.02? Assume a previous study showed that 60 out of 100 Columbia, MD households had
household income greater than $70,000.
3248 households (depending on how you rounded z*)
b. (6 points) Suppose you sample the number of households found in Part A, and you find
that 2110 of these households have a household income greater than $70,000. Use this
information to construct a 98% confidence interval for the population proportion of
Columbia, MD households that have a household income greater than $70,000 (use 3
decimal places for the endpoints of your interval).
(0.630,0.669)
c. (4 points) Explain what the 98% confidence interval (found in Part B) means in the
context of this problem.
We can be 98% confident that the true population proportion of Columbia, MD
households with household income greater than $70,000 is contained in the above
interval.
7. The following contingency table shows some demographic characteristics of the
10,081 Fall 2011 HCC credit students:
Day Students
Evening Students
Total
Full-Time
Students
3,483
230
3,713
Part-Time
Students
4,456
1,912
6,368
Total
7,939
2,142
10,081
Suppose you want to estimate (via a confidence interval) the proportion of fall 2011 HCC
credit students who have violated the new campus smoking ban. Each of the following
two sampling scenarios violates one assumption/condition that is necessary to estimate a
valid confidence interval. For parts A and B below, identify the assumption/condition that
is violated (your choices include independence assumption, randomization condition,
10% condition, nearly normal condition, counted data condition, and expected cell
frequency condition).
a. (3 points) You stratify the students into day and evening. You then sample 500 of the
day students using a simple random sample.
Randomization condition
b. (3 points) You use a simple random sample to sample 2,500 students.
10% condition
8. A gynecologist is about to interpret a cervical cancer screening test that she just
administered to one of her patients. Her hypotheses are as follows:
H0: The test results are “negative” and treatment is not necessary
HA: The test results are “positive” and aggressive treatments (surgery and radiation
therapy) are recommended
a. (5 points) What is a Type I error in the context of this problem, and what are its
consequences?
The null hypothesis is incorrectly rejected. This will lead to unnecessary and
potentially risky medical procedures.
b. (5 points) What is a Type II error in the context of this problem, and what are its
consequences?
The null hypothesis is incorrectly not rejected (i.e. it should be rejected, but it’s not).
This will lead to the cervical cancer being untreated.
9. (4 points each) Suppose the distribution of the number of Facebook friends that
Americans ages 18 to 24 have is normally distributed with mean 75 and standard
deviation 15. Suppose you are going to sample 64 Americans (ages 18 to 24) and
calculate the sample average ( Y ) number of Facebook friends. Find the following (round
the probabilities to 3 decimal places):
a. What is the sampling distribution of the sample means having sample size 64?
N( 75, 15/8)
b. Find the value of a cut off (y) such that the probability that Y (the sample average) is
greater than that cut off is 0.20 (in other words, find the value of y such that:
P( Y >y)=0.20) (round to the nearest tenth)
76.6 friends
c. Find the probability that a given person in your sample has fewer than 85 Facebook
friends. (round to 3 decimal places
0.748
d. Find the probability that the sample average ( Y ) is 80 facebook friends or more.
(round to 3 decimal places)
0.004