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Transcript
Math 155 Practice Final Exam Questions
1. For a class of 140 students, scores on a recent exam (on a scale from 0 to 100) were as
follows:
Score
90-100 80-89 70-79 60-69
Relative Frequency
.15
.25
.35
.10
What is the frequency of a score between 0 and 59?
a.
b.
c.
d.
e.
.15
.85
0
21
30
2. The statistics π‘₯Μ… = 25, 𝑠 2 = 16 are computed from sample data from a population.
What is the z-score for an observation: π‘₯ = 9?
a.
b.
c.
d.
e.
1
-1
4
-4
None of the above
3. For a sample of 25 observations, the sum of all observations is 455 and the sum of all
squared observations is 16,057.
The sample standard deviation is equal to which of the following?
a.
b.
c.
d.
e.
16
18
25.6
256
324
4. If a balanced die is rolled twice, the probability that both rolls are the same is equal to
which of the following?
a.
b.
c.
d.
e.
1/36
2/36
1/6
2/6
None of the above
The box plots below are for the reaction time (in seconds) from a study of 100
subjects where 50 were subjected to a non-threatening stimulus (NT) and the other
50 were subjected to a threatening stimulus (T).
Boxplot of TIME
NT
T
2.6
2.4
TIME
2.2
2.0
1.8
1.6
1.4
1.2
Panel variable: STIMULUS
Questions 5 and 6 below refer to the box-plots above.
5. What percentage of the subjects who received a threatening stimulus had reaction
times exceeding 1.8 seconds?
a.
b.
c.
d.
e.
0%
25%
50%
75%
100%
6. Which of the following statements is incorrect?
a. The inter-quartile range for the group who received a non-threatening stimulus
was .3 seconds.
b. The twenty-fifth percentile of reaction times for those who received a nonthreatening stimulus was greater than the seventy-fifth percentile for those who
received a threatening stimulus.
c. The difference in median reaction times for the two groups is .3 seconds.
d. In all of the data there was only one outlier.
e. The reaction times for the 50 subjects who received the threatening stimulus were
from a low of 1.3 seconds to a high of 2.1 seconds.
Questions 7-9 refer to the stem and leaf diagram below.
The stem and leaf diagram below is for EPA gas mileage (MPG) data for a sample of 80 Honda
CRV’s (model year 2008) that were tested for fuel efficiency. In this figure, a mileage of 30.0
would have a stem of 30 and a leaf of 0 (representing the fractional part of a mile per gallon).
You are also given for this data the following summary statistics:
βˆ‘80
𝑖=1 π‘₯𝑖 = 2,968.3
Stem-and-leaf of MPG
Leaf Unit = 0.10
1
2
5
9
13
21
38
(18)
24
16
11
6
3
2
1
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
N
and
2
βˆ‘80
𝑖=1 π‘₯𝑖 = 109,794
= 80
0
8
579
1269
2489
13566789
00123344556677889
001112233445667789
22345678
00347
02557
002
1
2
9
7. Determine the median mileage for this sample.
a.
b.
c.
d.
e.
36.95
37.00
37.05
37.10
37.15
8. Determine the mean mileage to the nearest .01 mile per gallon for this sample.
a.
b.
c.
d.
e.
29.68
36.95
37.00
37.05
37.10
9. Determine the percentage of cars in the sample whose MPG rating met or exceeded 39
miles per gallon.
a.
b.
c.
d.
e.
14%
16%
18%
20%
22%
10. In a gambling game, two different numbers are selected at random from the whole
numbers 1, 2, 3, … , 10 to form the β€œwinning” combination. You play the combination 1
and 3.
The probability that you will win is:
a.
b.
c.
d.
e.
1/10
4/10
1/90
1/45
1/100
Use the following information for questions 11-13.
The outcome for rolling a pair of balanced dice is the sum of the dots on the upward
faces. The probability model for the different outcomes is:
Sum
2
3
4
5
6
7
8
9
10
11
12
Probability 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
Events: 𝐴 = {7 ≀ sum ≀ 12} , 𝐡 = {6 ≀ sum ≀ 8}
11. Determine the probability that event A will not occur.
a.
b.
c.
d.
e.
5/12
5/11
6/11
6/12
7/12
12. Determine the probability that event A occurs, given that event B occurs.
a.
b.
c.
d.
e.
11/16
11/21
11/36
16/36
21/36
13. Determine the probability that event A occurs, or event B occurs, or both events occur.
a.
b.
c.
d.
e.
11/36
15/36
21/36
26/36
37/36
14. An experiment has 3 possible outcomes denoted by E1 , E2 , E3 . Which of the following is
the probability missing in the table below?
a.
b.
c.
d.
e.
Outcome
E1
E2
Probability
.24
.33
E3
.33
33
.43
43
It cannot be determined without additional information.
15. The probability that event E occurs, or event F occurs, or that both E and F occur is .60.
The probability that event E occurs is .48. The probability that event F occurs is .57.
Determine the probability that both events E and F occur.
a.
b.
c.
d.
e.
.03
.09
.12
.45
1.05
16. 𝑋 is a discrete random variable with p  1 ο€½ .6 , p  2  ο€½.3, p  3 ο€½.1 .
The mean value of 𝑋 is _____.
a.
b.
c.
d.
e.
1.0
1.5
2.0
2.7
3.0
17. 𝑋 is a discrete random variable with p  1 ο€½ .6 , p  2  ο€½.3, p  3 ο€½.1 .
The standard deviation of 𝑋 is _____.
a.
b.
c.
d.
e.
0.450
0.671
1.000
1.643
2.700
18. A fair coin is flipped 5 times. Determine the probability that at least 4 heads occur in the
5 flips.
a.
b.
c.
d.
e.
1/32
4/5
6/32
26/32
31/32
19. For an adult population, the average weight is 188 pounds and the standard deviation is
12 pounds. The weights for this population are normally distributed. You randomly
select 4 of these adults and compute the average value: 𝑋̅.
Determine the probability that 𝑋̅ exceeds 194.
a.
b.
c.
d.
e.
.1587
.1915
.3085
.3413
Cannot be determined.
20. A balanced die is rolled 20 times. On average, the number of β€œsixes” that you roll is
equal to _____.
a.
b.
c.
d.
e.
3
10/3
6
10
Cannot be determined since the number of sixes rolled is random.
21. An unbalanced die is manufactured so that there is a 20% chance of rolling a β€œsix.” The
die is rolled twenty times. The probability of rolling at least 4 β€œsixes” is equal to ______.
a.
b.
c.
d.
e.
.370
.411
.589
.630
.800
22. If X is normally distributed with mean 84 and variance 9, then 𝑃(𝑋 < 75) = _____.
a.
b.
c.
d.
e.
.0001
.0013
.1587
.3413
.9999
23. The statistics π‘₯Μ… = 25, 𝑠 2 = 16 are computed from sample data from a population.
What observation π‘₯ corresponds to a z-score -.5?
a.
b.
c.
d.
e.
17
19
21
23
27
24. Thirty subjects are selected at random from a population of adult males. In this group,
20% of the population is at least 40% over their β€œdesirable” weight. 𝑋 is the number of
males in this sample who are at least 40% over their β€œdesirable” weight.
Which of the following describes the probability distribution of the random variable 𝑋?
a.
b.
c.
d.
e.
𝑿 is approximately normal with mean 6.
𝑿 is approximately normal with mean 12.
𝑿 is binomial with mean 6.
𝑿 is binomial with mean 12.
None of the above statements is true.
25. Two hundred and fifty-six subjects are randomly selected from a population of adult
males. The average weight for this population is 192 pounds and the standard deviation
is 16 pounds. 𝑋̅ is the average weight for this sample.
Which of the following statements is correct?
a. The random variable
normal.
b. The random variable
normal.
c. The random variable
normal.
d. The random variable
normal.
e. The random variable
normal.
Μ… βˆ’ πŸπŸ—πŸ
𝑿
πŸπŸ”
has a distribution that is approximately standard
Μ… βˆ’ πŸπŸ—πŸ
𝑿
πŸ’
has a distribution that is approximately standard
Μ… βˆ’ πŸπŸ—πŸ
𝑿
𝟏
has a distribution that is approximately standard
Μ… βˆ’ πŸπŸ—πŸ
𝑿
𝟏/πŸ’
has a distribution that is approximately standard
Μ… βˆ’ πŸπŸ—πŸ
𝑿
𝟏/πŸπŸ”
has a distribution that is approximately standard
26. The random variable X has mean 72 and variance 36.
For a sample of size n ο€½ 81 from this population, calculate the z-score of a sample mean
equal to 73.
a.
b.
c.
d.
e.
.17
.67
1.00
1.33
1.50
27. Which statement regarding the sampling distribution of the sample mean 𝑋̅ is incorrect?
a. On average, the sample mean is equal to the population mean.
b. The standard deviation of the sample mean is equal to the standard deviation of
the population divided by the sample size.
c. The distribution of the sample mean is approximately normal if the sample size is
sufficiently large.
d. The t-distribution is not appropriate for approximating the distribution of the
sample mean.
e. The mean and variance of the sample mean can be determined from the mean and
variance of the population together with the sample size.
28. For a random variable 𝑋, the standard deviation is πœŽπ‘‹ = 10.
How large a random sample is required to make the standard error of the mean equal to
.1?
a.
b.
c.
d.
e.
10
100
1,000
10,000
100,000
29. Which of the following statements regarding the large-sample (i.e. 𝑛 β‰₯ 30), 95%
confidence interval for a population mean is incorrect?
a. If the sample size is quadrupled, then the width of the interval is halved.
b. The width of the interval is 1.96 times the standard error of the mean.
c. The midpoint of the interval is the sample mean, a point estimator of the
population mean.
d. In repeated use of this procedure, there is a 95% chance that the true mean will be
contained in the confidence interval
e. The value of 𝜢 is .05.
30. A small sample t-test is conducted to test the null hypothesis πœ‡ = 30 against the
alternative that the population mean exceeds 30. The population is normally
distributed. For a sample of size 16 the value of the test statistic is 2.125.
Which of the following statements is correct?
a. The null hypothesis is rejected at a 20% level of significance, but it is not rejected
at a 10% level of significance.
b. The null hypothesis is rejected at a 10% level of significance, but it is not rejected
at a 5% level of significance.
c. The null hypothesis is rejected at a 5% level of significance, but it is not rejected at
a 2.5% level of significance.
d. The null hypothesis is rejected at a 2.5% level of significance, but it is not rejected
at a 1% level of significance.
e. The null hypothesis is rejected at a 1 % level of significance, but it is not rejected at
a .5% level of significance.
31. For a sample of size 16 randomly selected from a normal population, the sample
standard deviation is 8.
Determine the sampling error with a confidence level of 98%.
a.
b.
c.
d.
e.
1.301
2.583
2.602
5.166
5.204
32. For a large sample, 2-sided, Z-test about a population mean, the p-value associated with
the statistic value 𝑍 = 1.54 is _________.
a.
b.
c.
d.
e.
.0618
.0813
.1236
.4382
.8764
33. For a large sample, 1-sided, Z-test about a population mean where the alternative
hypothesis is: π»π‘Ž : πœ‡π‘‹ < πœ‡0 , the p-value associated with the statistic value 𝑍 = βˆ’2.26 is
______.
a.
b.
c.
d.
e.
-.0119
.0119
.0122
-.4881
.4881
34. From sample data for a sample of size 50, a 97% confidence interval for a population
mean is determined to be equal to (54.2 , 57.6).
Which of the following statements is false?
a. The probability that the interval (54.2 , 57.6) contains the true population mean
is .97.
b. The value of the sample mean for this set of data is 55.9.
c. The sampling error is 1.7.
d. We are highly confident that the population mean is at most 57.6.
e. The π’›πœΆ/𝟐 value used in computing this interval is 2.170.
35. For any hypothesis test, which of the following are true?
I.
II.
III.
a.
b.
c.
d.
e.
The level of significance is equal to the probability of a type I error.
The level of significance is equal to the probability that the null hypothesis is
accepted when in fact the alternative hypothesis is true.
The level of significance is equal to the probability that the test statistic falls in
the rejection region.
Only statement I is true.
Only statement II is true.
Only statement III is true.
Only statements I and II are true.
Only statements I and III are true.
36. Which of the following statements concerning the probability that a random variable 𝑋
falls within the interval πœ‡π‘‹ ± 1.50πœŽπ‘‹ is correct?
a. The probability is equal to 𝑷(βˆ’πŸ. πŸ“πŸŽ ≀ 𝒁 ≀ 𝟏. πŸ“πŸŽ) where 𝒁 is a standardnormal, random variable.
b. The probability is midway between .68 and .95.
c. The probability is between 4/9 and 5/9.
d. The probability is midway between 5/9 and 3/4.
e. The probability is at least 5/9.
37. A 95% confidence interval for the proportion of voters favoring a certain ballot
measure is computed as (. 48402 , .54598) for a sample of 1,000 voters.
Determine the number of voters in the sample who favored this ballot proposition.
a.
b.
c.
d.
e.
510
515
520
525
530
38. A 95% confidence interval for the proportion of voters favoring a certain ballot
measure is computed as (. 48904 , .55096) from a sample of 1,000 voters.
Which of the following statements is correct?
a. It has been proven that more than 50% of voters are in favor of this ballot
measure.
b. With 95% confidence, we know that at least 50% of the voters are in favor of this
ballot measure.
c. There is a 5% chance that fewer than 50% of the voters favor this ballot measure
d. There is insufficient evidence to be 95% confident that more than 50% of the
voters favor this ballot measure.
e. None of the above is correct.
39. Which of the following is the rejection region for a 5% level, t- test of the following type
relying on a sample of size 28 from a normal population?
𝐻0 : πœ‡ = 26 , π»π‘Ž : πœ‡ < 26
a.
b.
c.
d.
e.
|𝒕| < 2.052
𝒕 < 2.052
𝒕 < βˆ’2.052
|𝒕| > 1.703
𝒕 < βˆ’1.703
40. Which of the following is the p-value for a test of the following type relying on a sample
of size 28 from a normal population where the sample data results in
𝑋̅ βˆ’26
𝑠/√28
= βˆ’2.473?
𝐻0 : πœ‡ = 26 , π»π‘Ž : πœ‡ < 26
a.
b.
c.
d.
e.
.005
.010
.020
.025
.050
41. For a one-sided test concerning a population proportion, the p-value associated with
the test statistic computed from the data is .0260.
Which of the following statements is false?
a.
b.
c.
d.
We would fail to reject the null hypothesis at a level of significance of 2%.
For a 5% level of significance, the test statistic would be in the rejection region.
For a 2-sided test, the same data would have resulted in a p-value of .0520.
For a 2-sided test, the same data would have resulted in rejecting the null
hypothesis at a 5% level of significance.
42. Which of the following statements regarding the 100(1 βˆ’ 𝛼)% confidence interval
π‘₯Μ… ± 𝑧𝛼/2 πœŽπ‘‹ /βˆšπ‘› for the unknown mean of a population is false?
a. The sampling error increases if the level of confidence is increased while the
sample size remains the same.
b. The sampling error decreases if the level of confidence remains the same while
the sample size is increased.
c. The midpoint of the interval is always equal to the population mean.
Μ….
d. The ratio πˆπ‘Ώ /βˆšπ’ is equal to the standard deviation of the point estimator 𝒙
e. If the confidence level is increased, then π’›πœΆ/𝟐 is also increased.
43. Which of the following statements is false?
a. Statistics are used as point estimators of population parameters.
b. A 2-sided hypothesis test could be used to detect whether the mean weight of 60
year old males in the US population today has shifted from what it was 2 decades
ago.
c. Properties of the sampling distribution of a statistic are not needed in creating
confidence interval formulas.
d. When a large sample z-test results in the null hypothesis 𝝁 = πŸπŸ‘. 𝟐 being
rejected, it has not been proven that 𝝁 β‰  πŸπŸ‘. 𝟐.
e. A matched pairs statistical study results in a 95% confidence interval for 𝝁𝟏 βˆ’ 𝝁𝟐
given by (βˆ’πŸ. πŸ— , 𝟐. πŸ•). There is no conclusive evidence that the means are not
equal.
44. Which of the following statements is true?
Μ… βˆ’ πŸπ’” , 𝒙
Μ… + πŸπ’”).
a. For any sample, 75% of the data lies in the interval (𝒙
b. For any mounded and symmetric data set, 95% of the data lies in the interval
(𝒙
Μ… βˆ’ πŸπ’” , 𝒙
Μ… + πŸπ’”).
Μ… βˆ’ πŸ‘π’” , 𝒙
Μ… + πŸ‘π’”).
c. For any sample, 88.9% of the data lies in the interval (𝒙
d. For any mounded and symmetric data set, 99.7% of the data lies in the interval
(𝒙
Μ… βˆ’ πŸ‘π’” , 𝒙
Μ… + πŸ‘π’”).
e. The conclusions of Chebychev’s rule are always valid, no matter the shape of the
population being sampled.
45. Which of the following statements is false?
a. For a given 2-sample data set with independent sampling, the conclusion
regarding the equality of the two distributions reached in applying the Wilcoxson
rank sum test is the same as the conclusion reached using the Mann-Whitney Utest.
b. The advantage of a matched-pairs experimental design is to reduce variance in
the test statistic in contrast with the independent sample design.
c. Smaller variance in a test statistic lends greater precision to the conclusions of the
test.
d. Other than the difference in sample size, the validity requirements for a small
sample t-test (𝒏 < 30) about a population mean are the same as those for a large
sample z-test (𝒏 β‰₯ πŸ‘πŸŽ) about a population mean.
e. If two population means are truly different, then it becomes easier to detect as
the sample size increases.
Use the following information for questions 46-49.
At the 59,600 student campus of ESU (Enormous State University) a random sample of
500 students was selected to study academic probation by gender. The results were:
Probation
No Probation
Male
15
205
Female
20
260
46. Determine the value of the point estimator for the proportion of all students on
probation.
a.
b.
c.
d.
e.
.03
.04
.07
.08
.35
47. Determine the value of the point estimator for the proportion of all female students on
probation.
a.
b.
c.
d.
e.
.020
.040
.071
.077
.142
Use the following information for questions 46-49.
At the 59,600 student campus of ESU (Enormous State University) a random sample of
500 students was selected to study academic probation by gender. The results were:
Probation
No Probation
Male
15
205
Female
20
260
48. A student is selected at random from this sample. Given that the student is on
probation, calculate the probability that the student is a female.
a.
b.
c.
d.
e.
.07
.43
.50
.57
.64
49. Determine the value of the point estimator for the proportion of all students who are
male.
a.
b.
c.
d.
e.
.11
.22
.33
.44
.55
Use the following information for questions 50-51.
For a discrete random variable 𝑋, you are given the following information:
π‘₯
𝑝(π‘₯)
0
.01
1
.04
2
.15
3
.40
7
4
.18
5
.12
6
.07
7
.03
7
βˆ‘ π‘₯ 𝑝(π‘₯) = 3.49 ,
βˆ‘ π‘₯ 2 𝑝(π‘₯) = 14.11
π‘₯=0
π‘₯=0
50. Determine the standard deviation for this population.
a.
b.
c.
d.
e.
1.14
1.39
1.93
3.76
14.11
51. Determine the probability that the random variable 𝑋 is within one standard deviation
of its mean.
a.
b.
c.
d.
e.
.40
.55
.58
.68
.73
------------------------------------------------------------------------------------------------------------------------------52. The probability that the first ball drawn from an urn is red is equal to .44. Given that
the first ball drawn is red, the probability that the next ball drawn is blue is equal to .25.
You play a game where you win if the first ball you draw is red and the second ball that
you draw is blue.
Determine the probability that you will not win the game.
a.
b.
c.
d.
e.
.110
.190
.345
.667
.890