Practice Exam 3

Transcription

Practice Exam 3

Practice Exam 3
1. A simple random sample of commuting UF students was taken to determine the
average amount of money spent on gas per week. The 34 students surveyed gave an
average expenditure of $15.80 with a standard deviation of $4.25. Does it appear that
the population average gas expenditure per week is greater than $17.50?
(i) What type of problem is this?
(ii) Define the parameter.
(iii) Are the assumptions met?
(iv)State the null and the alternative hypothesis.
(v) Calculate the test statistic.
(vi) Find the p-value and interpret.
2. A national study estimated that in 2005, college students spent an average of 2 hours
per day online. An SRS of 150 UF students produces an average of 1.8 hours. The
standard deviation is 0 .4. Do UF students spend less time than average online?
3. A company tries to fill their canned drinks to 12 fl. oz. They control this by taking a
SRS of 6 cans every 30 minutes. The data is below. Is the average level of fill in the
drinks significantly different from what it should be?
11.2
11.3
11.4 11.4 10.9 10.7
4. Is the percentage of college students looking for a job higher than the percentage
looking for a job in the general population? A survey reported that at any given time,
19% of Americans are looking for a job. A survey of 200 college students was conducted
and 60 reported looking for a job.
5. The GSS asked participants whether they agree with the statement that a marriage
without children isn’t fully complete. In this survey they found out that 623 agreed or
strongly agreed with the statement, while 731 did not. Is the population proportion of
Americans that agrees or strongly agrees different from 83%?
6. Is the percentage of people who call themselves very religious less than 25%? The
1998 GSS reported that 268 of 1427 people considered themselves very religious. Is
there evidence to show that the percentage is significantly less than 25%?
7. The GSS asked 1594 people if they had been to an amateur or professional sports
event within the past twelve months. Out of 1594 people, 855 people reported that they
had. Is there evidence to show that the population proportion of Americans who have
been to a sports event in the past twelve months is different from 0.50?
8.An inspector inspects large truckloads of potatoes to determine the proportion in the
shipment with major defects prior to using the potatoes to make chips. If there is clear
evidence that this proportion is less than 0.10, she will accept the shipment. To do so, she
selects a simple random sample of 200 potatoes from the more than 3000 potatoes on the
truck. Only 8 of the potatoes sampled are found to have major defects. Does she accept
the shipment?
9. A newsletter recently reported that 90% of adults drink milk. A regional farmers’
organization is planning a new marketing campaign across its tri-county area. They
randomly poll 600 people in the area. In this sample, 525 people said that they drink
milk. Do these data provide strong evidence that the 90% figure is not accurate for this
region?
For questions 10-19, answer the questions for each problem.
10. A high school biology teacher thinks that a new hands-on method of teaching will
improve student scores on an end of the year test. She teaches 10 students the old way,
and she teaches 7 students the new hands-on way. Are the test scores for the students that
learned the new way higher than those that learned the old way?
New Method
Old Method
new
old
N
7
10
Mean
87.7
83.50
88 78
65 86
StDev
11.6
9.58
95 96 100
92 95 84
67 90
77 96
83 75 82
SE Mean
4.4
3.0
Difference = mu (new) - mu (old)
Estimate for difference: 4.21429
95% CI for difference:
(-7.48721, 15.91578)
Boxplot of new, old
100
Data
90
80
70
60
new
old
a.) What type of problem is this?
b.) What are the assumptions of the confidence interval? Are they satisfied?
c.) What is the 95% confidence interval? _________________________
d.) What is your conclusion?
e.) What is the short conclusion?
11. Each of a random sample of ten college freshman takes a mathematics aptitude
test both before and after undergoing an intensive training course designed to
improve such test scores. Then, results of the students’ scores before and after the
course are listed below. Using the below results and the computer output below to
answer the following questions.
Student 1 Before 60 After 70 2 73 80 3 42 40 4 88 94 5 66 79 6 77 86 7 90 93 8 63 71 9 55 70 10 96 97 Paired T-Test and CI: before, after
Paired T for before - after
before
after
Difference
N
10
10
10
Mean
71.00
78.00
-7.00
StDev
17.07
16.77
5.25
SE Mean
5.40
5.30
1.66
95% CI for mean difference: (-10.76, -3.24)
T-Test of mean difference = 0 (vs not = 0): T-Value = -4.22
P-Value = 0.002
a.) What type of test is this?
c.) What is the 95% confidence interval? _______________
d.) What is the short conclusion?
e.) What is the full interpretation?
12. It is widely believed that those ages 16-24 feel differently than those 25 or older
about the legalization of marijuana. A national survey, which randomly selected from all
across the United States, found the following results:
Yes-Legalize
503
525
Ages 16-24
Age >25
No-Don’t Legalize
1021
1263
Total
1524
1788
Does it appear that the younger group feels differently from the older group about the
legalization of marijuana?
Sample
1
2
X
503
525
N
1524
1788
Sample p
0.330052
0.293624
Difference = p (1) - p (2)
95% CI for difference: (0.00475862, 0.0680980)
c.) What is the 95% confidence interval? _________________________
13. In 2010, the General Social Survey included a question that asked males and females
if they thought that Antarctic penguins were threaten. The possible responses were “A
great deal” and “not at all”.
Males (group 1)
Females (group 2)
A great deal
121
213
Not at all
30
17
Total
151
230
Make a 95% confidence interval for the difference in the population proportion between
men and women that feel that penguins in Antarctica are threatened a great deal.
Test and CI for Two Proportions
Sample
1
2
X
121
213
N
151
230
Sample p
0.801325
0.926087
Estimate for difference: -0.124762
95% CI for difference: (-0.196828, -0.0526971)
Test for difference = 0 (vs not = 0): Z = -3.39
P-Value = 0.001
Fisher's exact test: P-Value = 0.000
c.) What is the 95% confidence interval? ________________________________
e.) What is your quick interpretation?
14. A health professional believes that a new diet will increase average energy levels.
Eight randomly selected women are given a survey to measure their average energy
levels before they begin the diet. After 3 months, they are again given the survey.
Higher numbers relate to higher energy levels. The results are as follows:
Before
After
15
18
Before
After
Difference
N
8
8
8
22
23
25
23
18
22
24
25
Mean
20.5000
21.6250
-1.12500
StDev
4.7509
3.6228
2.41646
SE Mean
1.6797
1.2809
0.85435
14
18
19
17
95% CI for mean difference: (-3.14521, 0.89521)
Histogram of Differences
(with Ho and 95% t-confidence interval for the mean)
2.0
Frequency
1.5
1.0
0.5
0.0
_
X
Ho
-4
-3
-2
-1
Differences
0
1
2
c.) What is the 95% confidence interval? ________________________
e.) What is the quick interpretation?
27
27
15. Do UF students from out of state spend significantly less time at their parents’ house
(homes) than those from in state? A SRS found that out-of-state students spend an
average of 3.22 days per semester (std. dev. = 2.49) at their parents’ house (homes)
while in-state students spend an average of 5.89 days per semester (std. dev.=4.23) at
their parents’ house (homes).
Out of state
Instate
N
9
9
Mean
3.22
5.89
StDev
2.49
4.23
SE Mean
0.83
1.4
Difference = mu (Out of state) - mu (Instate)
95% CI for difference: (-6.22877, 0.89544)
T-Test of difference = 0 (vs not =): T-Value = -1.63 P-Value = 0.129 DF = 12
T-Test of difference = 0 (vs <): T-Value = -1.63 P-Value = 0.064 DF = 12
T-Test of difference = 0 (vs >): T-Value = -1.63 P-Value = 0.936 DF = 12
Boxplot of Out of state, Instate
14
12
Data
10
8
6
4
2
0
Out of state
Instate
b.) Are the assumptions met?
c.) What is the null hypothesis? ______________________________
d.) What is the alternative hypothesis? _____________
e.) Look at the output below. What is the corresponding p-value for this hypothesis?
____________________
f.) Write a conclusion.
16. A high school biology teacher thinks that a new hands-on method of teaching will
improve student scores on an end of the year test. She teaches 10 students the old way,
and she teaches 7 students the new hands-on way. Are the test scores for the students that
learned the new way higher than those that learned the old way?
New Method (1) 88 78 95 96 100 67 90
77 96 83 75 82
Old Method (2) 65 86 92 95 84
new
old
N
7
10
Mean
87.7
83.50
StDev
11.6
9.58
SE Mean
4.4
3.0
Difference = mu (new) - mu (old)
T-Test of difference = 0 (vs <): T-Value = 0.79
P-Value = 0.778
T-Test of difference = 0 (vs not =): T-Value = 0.79
T-Test of difference = 0 (vs >): T-Value = 0.79
DF = 11
P-Value = 0.445
P-Value = 0.222
DF = 11
DF = 11
b.) What is the null hypothesis? ______________
c.) What is the alternative hypothesis? _____________
d.) Look at the output below. What is the corresponding p-value for this hypothesis?
___________
e.) Write a conclusion.
17. It is widely believed that those ages 16-24 feel differently than those 25 or older
about the legalization of marijuana. A national survey found the results below. Does it
appear that the younger group feels differently from the older group about the
legalization of marijuana?
Yes-Legalize
No-Don’t Legalize Total
503
1021
1524
Ages 16-24
525
1263
1788
Age >25
Sample
1
2
X
503
525
N
1524
1788
Sample p
0.330052
0.293624
Test for difference = 0 (vs not = 0):
Z = 2.25
P-Value = 0.024
Test for difference = 0 (vs < 0):
Z = 2.25
P-Value = 0.988
Test for difference = 0 (vs > 0):
Z = 2.25
P-Value = 0.012
b.) What is the null hypothesis? _______________
_________
18. Each of a random sample of ten college freshman takes a mathematics aptitude
test both before and after undergoing an intensive training course designed to
improve such test scores. Then, results of the students’ scores before and after the
course are listed below. Using the below results and the computer output below to
answer the following questions. Is there evidence to show that the intensive training
course helped?
Student 1 Before 60 After 70 2 73 80 3 42 40 4 88 94 5 66 79 6 77 86 7 90 93 8 63 71 9 55 70 10 96 97 Paired T-Test and CI: before, after
Paired T for before - after
before
after
Difference
95% CI
T-Test
T-Test
T-Test
N
10
10
10
Mean
71.00
78.00
-7.00
StDev
17.07
16.77
5.25
for mean difference:
of mean difference =
SE Mean
5.40
5.30
1.66
(-10.76, -3.24)
0 (vs not = 0): T-Value = -4.22 P-Value = 0.002
0 (vs < 0): T-Value = -4.22 P-Value = 0.001
0 (vs > 0): T-Value = -4.22 P-Value = 0.999
a) What type of problem is this?
b) What is the null hypothesis? _______________
c) What is the alternative hypothesis? _____________
d) Look at the output below. What is the corresponding p-value for this hypothesis?
___________
e) Write a conclusion.
19. In 2010, the General Social Survey included a question that asked males and females
if they thought that Antarctic penguins were threaten. The possible responses were “A
great deal” and “not at all”.
Males (group 1)
Females (group 2)
A great deal
121
213
Not at all
30
17
Total
151
230
Make a 95% confidence interval for the difference in the population proportion between
men and women that feel that penguins in Antarctica are threatened a great deal.
Test and CI for Two Proportions
Sample
1
2
X
121
213
N
151
230
Sample p
0.801325
0.926087
95% CI for difference: (-0.196828, -0.0526971)
Test for difference = 0 (vs not = 0): Z = -3.39 P-Value = 0.001
Test for difference = 0 (vs < 0): Z = -3.39 P-Value = 0.000
Test for difference = 0 (vs > 0): Z = -3.39 P-Value = 1.000
___________
20. A health professional believes that a new diet will increase average energy levels.
Eight women are given a survey to measure their average energy levels before they begin
the diet. After 3 months, they are again given the survey. Higher numbers relate to
higher energy levels. The results are as follows:
22
25
18
24
14
19
27
Before 15
18
23
23
22
25
18
17
27
After
Before
After
Difference
N
8
8
8
Mean
20.5000
21.6250
-1.12500
StDev
4.7509
3.6228
2.41646
SE Mean
1.6797
1.2809
0.85435
T-Test of mean difference = 0 (vs not = 0): T-Value = -1.32
P-Value = 0.229
T-Test of mean difference = 0 (vs < 0): T-Value = -1.32
P-Value = 0.115
T-Test of mean difference = 0 (vs > 0): T-Value = -1.32
P-Value = 0.885
_________
21. Do UF students from out of state spend significantly less time at their parents’ house
(homes) than those from in state? A survey found that out-of-state students spend an
average of 3.22 days per semester (std. dev. = 2.49) at their parents’ house (homes)
while in-state students spend an average of 5.89 days per semester (std. dev.=4.23) at
their parents’ house (homes).
Out of state
Instate
N
9
9
Mean
3.22
5.89
StDev
2.49
4.23
SE Mean
0.83
1.4
Difference = mu (Out of state) - mu (Instate)
T-Test of difference = 0 (vs not =): T-Value = -1.63
P-Value = 0.129
DF = 12
T-Test of difference = 0 (vs <): T-Value = -1.63
P-Value = 0.064
DF = 12
T-Test of difference = 0 (vs >): T-Value = -1.63
P-Value = 0.936
DF = 12
a) What type of problem is this?
b) What is the null hypothesis? _______________
c) What is the alternative hypothesis? _____________
d) Look at the output below. What is the corresponding p-value for this hypothesis?
___________
e) Write a conclusion.
Short Answer
22. The cost of hiring an employee (excluding salary) can range from about $1,500 for a
secretary to more than $40,000 for a manager. To estimate its mean cost of hiring an
entry-level secretary, a large corporation randomly selected eight of the entry-level
secretaries it had hired during the last two years and determined the costs (in dollar)
involved in hiring each. The following data were obtained:
2,100 1,650 2250
2,035 2,245 1,980 1,700 2,190
a) State the assumptions for a confidence interval for the population mean and discuss if
the assumptions are met.
b) Make a 95% CI for µ and interpret this interval.
c) Compute a 90% confidence interval for the population mean.
d) Which interval is wider and why?
23. A federal bank examiner is interested in estimating the mean outstanding principle
balance of all home mortgages foreclosed by the bank due to default by the borrower
during the last 3 years. A SRS of 12 foreclosed mortgages yielded the following data ( in
dollars):
95,982
81,422
39,888
46,836
66,899
62,331
105,812
55,545
56,635
72,123
69,110
59,200
a) State the assumptions for a confidence interval for the population mean and discuss if
the assumptions are met.
b) Construct a 95% CI for μ. Interpret the interval.
c) Construct a 99% confidence interval.
d) Which interval is wider and why?
24. We want to know μ, the true average temperature in Florida on a certain day. We
take a SRS of 10 cities and get the following:
75 78 84 86 80 92 85 84 90 88
i) What are the assumptions that need to be met for constructing a confidence interval?
ii) Can we construct a 99% confidence interval for μ?
iii) If so, construct one. If not, explain why not.
.
25. A survey asks 231 college students how many sexual partners they have had in their
lives. The sample mean was 4.641 and the sample standard deviation equals 6.33.
i) Is the parameter being estimated the population proportion or the population mean?
ii) Are the assumptions for a confidence interval for the population parameter met?
iii) Find the 95% confidence interval for the parameter.
iv) Interpret the interval.
26. If you increase the sample size of a confidence interval, what happens to the width of
a confidence interval? ____________________
27. A survey was conducted to University of Florida students after they returned from
Spring Break. Thirty-five students were asked how much money they spent during Spring
Break. A 95% confidence interval was created from the results. The interval was from
150 to 200 dollars.
For each of the following statements about the above confidence interval, determine if
they are correct or incorrect. If they are incorrect, state why.
1.) The 95% confidence interval for the average amount of money spent by all UF
students over Spring Break is between 150 and 200 dollars.
2.) 95% of all students spend between 150 and 200 dollars during Spring Break.
28. A 95% confidence interval for the mean number of televisions per American
household is (1.15, 4.20).
For each of the following statements about the above
confidence interval, determine if they are correct or incorrect. If they are incorrect, state
why.
a. The probability that  is between 1.15 and 4.20 is .95.
b. We are 95% confident that the true mean number of televisions per American
household is between 1.15 and 4.20.
c. 95% of all samples should have x-bars between 1.15 and 4.20.
d. 95% of all American households have between 1.15 and 4.20 televisions.
e. Of 100 intervals calculated the same way (95%), we expect 95 of them to capture
the population mean.
f. Of 100 intervals calculated the same way (95%), we expect 95 of them to capture
the sample mean.
29. The Harvard School of Public Health College Alcohol Study Survey surveys college
students in about 200 colleges in 1993, 1997, and 1999. They asked the students
demographic questions as well as questions about their drinking habits. They were
especially interested in the binging habits of college students. The survey defines an drink
as “12 oz bottle or can of beer, a 4 oz (120 mL ) glass of wine, a 12 oz. (360mL) bottle or
can of wine cooler, or a shot (1.25 oz or 37 mL) of liquor straight or in a mixed drink.”
Binge Drinking is considered drinking 5 drinks in a row for males and 4 drinks in a row
for females. This information is from the 2001 study in which a SRS of college students
was taken. Suppose that we want to find the population proportion of American male
college students that are binge drinkers. For 3925 males, 1908 were binge drinkers and
2017 were not binge drinkers.
a.) Are the assumptions met for the confidence interval for the population proportion of
male college students that binge drink?
b.) Make a 95% CI for p, the proportion of male college students that binge drink and
interpret.
Suppose that we want to determine the proportion of female American college students
that are binge drinkers. The same survey asked 6979 females. Out of 6979 females,
2854 were binge drinkers and 4125 were not binge drinkers.
c.) Are the assumptions met for the confidence interval for the population proportion of
female college students that binge drink?
d.) Make a 99% CI for p, the proportion of female college students that binge drink and
interpret. 30. A SRS of high school students was surveyed. Each student was asked about their
biological parents smoking habits and their own smoking habits. Suppose that we want to
estimate the population proportion of students that smoke given that at least one of the
parent’s smokes. Out of 4024 students, 816 students smoked and 3208 students did not
smoke.
a.) Are the assumptions met?
b.) Make a 99% CI for p, the proportion of students that smoke given that at least one
of the parents’ smokes.
31. You decide to survey people and ask them if they intended to spend or save their tax
refund. You want to be 99% confident in your answer and you want to have a margin of
error of 0.02. What size sample do you need if . . . .
a.) You have no clue what proportion of people will spend their tax refund
b.) You think that the proportion will be close to 0.80.
32. You decide to survey college students and ask them how much they spend on
entertainment per month. You want to be 95% confident in the answer and you want to
be within 2 dollars of the population mean. What size sample do you need if . . .
a.) All you know is that the amount spent on entertainment is typically between 0
dollars and 50 dollars a month.
b.) Last year, the standard deviation was 10 dollars.
33. What is the definition of a Type I error?
34. What is the definition of a Type II error?

Practice Exam 3

Transcription

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