Download Practice problems for the Final. Use the following to answer

Practice problems for the Final.
1. . The scores on a university examination are normally distributed with a mean of 62 and a standard
deviation of 11. If the bottom 5% of students will fail the course, what is the lowest mark that a student
can have and still be awarded a passing grade? [Answer: 44]
Use the following to answer questions 2-4. Suppose that a college determines the following
distribution for X = number of courses taken by a full-time student this semester.
Value of X
3
4
5
6
Probability
0.07
?
0.25
0.28
2. The probability for X = 4 is missing. What is it?
A) 0.07
B) 0.25
C) 0.40
D) 0.50 Answer: C
3. What is the average number of courses full-time students at this college take this semester?
A) 4 classes
B) 4.26 classes
C) 4.74 classes
D) 5 classes
Answer: C
4. What is the standard deviation of the number of courses full-time students at this college take this
semester?
A) 0.89 classes
B) 0.94 classes
C)1 class
D) 23.36 classes
Answer: B
Use the following to answer questions 5-8: If you draw an M&M candy at random from a bag of the
candies, the candy you draw will have one of six colors. The probability of drawing each color
depends on the proportion of each color among all candies made. Assume the table below gives the
probabilities for the color of a randomly chosen M&M.
Color
Prob
Brown Red
0.3
0.3
Yellow
?
Green Orange
0.1
0.1
Blue
0.1
5. What is the probability of drawing a yellow candy?
A) 0.1
B) 0.2
C) 0.3
D) Impossible to determine from the information
given. Answer: A
6. What is the probability of not drawing a red candy?
A) 0.3
B) 0.6
C) 0.7
D) 0.9 Answer: C
7. What is the probability that you draw neither a brown nor a green candy?
A) 0.3
B) 0.6
C) 0.7
D) 0.9 Answer: B
8. If you select two M&M’s and the colors are independent, then what is the probability that both are
the same color?
A) 0.01
B) 0.09
C) 0.22
D) 0.25 Answer: C
9. Suppose a fair coin is flipped twice and the number of heads is counted. Which of the following
is a valid probability model for the number of heads observed in two flips?
A) Number of Heads
Probability
0
¼
1
½
2
¼
B) Number of Heads
Probability
0
¼
1
¼
2
¼
C) Number of Heads
Probability
0
⅓
1
½
2
⅓
D) None of the above.
Answer: A
Use the following to answer questions 10-11: When figure skaters need to find a partner for “pair
figure skating,” it is important to find a partner who is compatible in weight. The weight of figure
skaters can be modeled by a normal distribution. For male skaters, the mean is 170 lbs. with a
standard deviation of 10 lbs. For female skaters, the mean is 110 lbs. with a standard deviation of 5
lbs. Let the random variable X = the weight of female skaters and the random variable Y = the
weight of male skaters.
10. What is P(X < 100)?
A) 0
B) 0.0228
C) 0.1587
D) 0.9772 Answer: B
11. Approximately 90% of the male skaters weigh more than how many pounds?
A) 157 lbs.
B) 163.5 lbs.
C) 176.5 lbs
D) 183 lbs.
Answer: A
Use the following to answer questions 12-14: In a large city, 72% of the people are known to own
a cell phone, 38% are known to own a pager, and 29% own both a cell phone and a pager.
12. What proportion of people in this large city own either a cell phone or a pager?
A) 0.29
B) 0.67
C) 0.81
D) 1.1 Answer: C
13. What is the probability that a randomly selected person from this city owns a pager, given that
the person owns a cell phone?
A) 0.266
B) 0.38
C) 0.403
D) 0.528 Answer: C
14. Are the events “owns a pager” and “owns a cell phone” independent?
A) Yes.
B) No, because P(owns a pager) and P(owns a cell phone) are not equal.
C) No, because P(owns a pager) and P(owns a pager | owns a cell phone) are not equal.
D) Cannot be determined.
Answer: C
Use the following to answer questions 15-18: The table below shows the political affiliation of 1000
randomly selected American voters and their positions on the school of choice program.
In Favor
Oppose
Democrat
260
40
Republican
120
240
Other
240
100
Let the event D = {voter is a Democrat}, R = {voter is a Republican}, and F = {voter favors the
school of choice program}. For each of the following questions, write the probability in symbols
(e.g., P(D)) and calculate the probability.
15 What is the probability that a randomly selected voter favors the school of choice program?
A) P(F) = 0.30
B) P(F) = 0.36
C) P(F) = 0.38
D) P(F) = 0.62
Answer: D
16. What is the probability that a randomly selected Republican favors the school of choice program?
A) P(F | R) = 0.12
B) P(R | F) = 0.19
C) P(F | R) = 0.33
D) P(R | F) = 0.36 Answer: C
17. What is the probability that a randomly selected voter who favors the school of choice program is
a Democrat?
A) P(D | F) = 0.26
C) P(F | D) = 0.48
Answer: B
B) P(D | F) = 0.42
D) P(F | D) = 0.87
18. A candidate thinks she has a good chance of gaining the votes of anyone who is a Democrat or
who is in favor of the school of choice program. What proportion of the 1000 voters is that?
A) P(D or F) = 0.26
B) P(D and F) = 0.65
C) P(D or F) = 0.66
D) P(D | F) = 0.92
Answer: C
19. Say True or False: In general, when the sample size stays the same, the higher the level
of confidence, the narrower the confidence interval. FALSE
20. Say True or False: The Central Limit Theorem states that as the sample size increases,
the distribution of the sample mean approaches a normal distribution with mean equal to
the population mean and standard deviation equal to the population standard deviation.
FALSE
21. What is the probability that a standard normal variable will be between -0.5 and 1.00?
0.5328
22. A mail-order computer business has five telephone lines. Let X denote the number of lines in
use at a specified time. Suppose the probability distribution of X is as given in the accompanying
table.
X
0
1
2
3
4
5
P(x)
.10
.15
.20
.25
.22
.08
Find the mean of X (mean = 2.58)
Use the Following for Questions 25 to 28: The time x (minutes) for a lab assistant to prepare the
equipment for a certain experiment is believed to have a uniform distribution over the interval a = 20
and b = 30; that is, its corresponding density curve has constant height over the interval [20, 30].
23. Write the probability density function (pdf) of x.
Answer: f(x) = 1/10 for all x in [20.30]
24. What is the probability that preparation time exceeds 27 minutes? Answer: 3/10
25. Find the preparation mean time Answer: 25
26. Use your answer to Question 25 to calculate the probability that preparation is within
2 minutes of the mean time? Answer: 4/10
27. A large retailer is studying the lead time (elapsed time between when an order is placed and when it
is filled) for a sample of recent orders. The lead times are reported in days.
Lead Time
0 to 5
6 to 10
11 to 15
16 to 20
21 to 25
Frequency
6
7
12
8
7
a) What proportion of orders are filled under 5 days? Answer: 6/40
b) What proportion of orders need more than 20 days? Answer: 7/40
28. The probability that I get job A is 0.45; the probability that I get job B is 0.60; and the probability that
I get both the jobs is 0.30. What is the probability that I get at least one job offer?Answer: 0.75
29. A manufacturer of TV sets wants to find the average selling price of a particular model. A random
sample of 35 different stores gives the mean sale price as $341 with a standard deviation of $16. Give a
90% confidence interval for the mean selling price of the TV model. What is the margin of error?
Answer: (336.55, 345.45), error = 4.45
30. Suppose an instructor gives an exam where the grades are normally distributed with mean 80 and
standard deviation 5.
a) Find the probability that a student will score at least 84 on this exam. Answer: 0.2118
b) Find the probability that a student will score between 75 and 85? Answer: 0.6826
c) This instructor wants to give those students in the top 2.5% an A on this exam. What will the cutoff
score be for an A? Answer: 89.79
d) If there are 10 students in the class, what is the probability that the average score will be less than
76? Answer: 0.0057
31. A company has a new project under way and selects five executives for a transfer from their current
jobs. A report had suggested that 75% of all executives in this company would like this new job.
a) What is the probability that at least three of these five selected like their new job? Answer: 0.8965
b) What is the probability that exactly three of the five selected like their new job? Answer: 0.2637
32. Thirty percent of the managers in a certain company have MBA degrees as well as professional
training. Eighty percent of all managers in the firm have professional training. If a manager is randomly
chosen and found to have professional training, what is the probability that he or she also has an MBA?
Answer: 0.375
33. If P(A) = .28, P( A
B) = .65, P(A
B) = .14, find P(B) Answer:0.51
34. The table below gives the values of a random variable X takes with the respective probabilities.
X
-1
0
1
P(X)
.1
.7
.2
a) Find the mean of X Answer:0.1
b) Find the standard deviation of X Answer:0.5385
From a group of six men and four women, a committee of four is to be chosen. What
is the probability that this committee consists of exactly two men and two women?
Answer:0.4286
35. A common test for AIDS is called the ELISA test. Among 1,000,000 people who are given the ELISA
test, we can expect results similar to those given in the following table:
Carry AIDS Virus
Do Not Carry AIDS Virus
Test is Positive
4885
73630
Test is Negative
115
921370
a) If one of these 1,000,000 people is selected randomly, what is the probability that the person carries
the AIDS virus? ANSWER 5000/1000000
b) What is the probability that the test is positive? ANSWER 78515/1000000.
c) What is the probability that the test is positive given that the person does not carry the AIDS virus?
ANSWER 73630/995000
36. A consumers research group sampled 100 hand-held video games, all of the same make and model.
The sample mean life (hours of operation before failure) was 560 hours. Assume the standard deviation
σ = 3.5 hours.
a) Find a 90% confidence interval estimate for the true mean life span of the video games
Answer: (559.42, 560.58)
b) What is the margin of error for a) Answer:0.58
c) Find a 96% confidence interval estimate for the true mean life span of the video games
Answer: (559.28, 560.72)
d) What is the margin of error for c)
Answer:0.72
37. One hundred volunteers who suffer from severe depression are available for a study. Fifty are
selected at random and are given a new drug that is thought to be particularly effective in treating
severe depression. The other 50 are given an existing drug for treating severe depression. A
psychiatrist evaluates the symptoms of all volunteers after four weeks in order to determine if there
has been substantial improvement in the severity of the depression. What is the explanatory variable
or factor in this study?
A) Which drug the volunteers receive.
B) The use of randomization and the fact that this was a comparative study.
C) The extent to which the depression was reduced.
D) The use of a psychiatrist to evaluate the severity of depression. Answer: A
38. Suppose volunteers were first divided by gender, and then half of the men were randomly
assigned to the new drug and half of the women were assigned to the new drug. The remaining
volunteers received the other drug. What is this an example of?
A) Replication.
B) Confounding. The effects of gender will be mixed up with the effects of the drugs.
C) A block design.
D) A matched-pairs design.
Answer: C
Use the following to answer questions 38-40. Suppose that a college determines the
following distribution for X = number of courses taken by a full-time student this semester.
Value of X
3
4
5
6
Probability
0.07
?
0.25
0.28
39. The probability for X = 4 is missing. What is it? A) 0.07 B) 0.25 C) 0.40 D) 0.50 Answer: C
40. What is the average number of courses full-time students at this college take this semester? A) 4
classes B) 4.26 classes C) 4.74 classes D) 5 classes Answer: C
41. What is the standard deviation of the number of courses full-time students at this college take this
semester? A) 0.89 classes B) 0.94 classes C)1 class D) 23.36 classes Answer: B
42. In order to assess the effects of exercise on reducing cholesterol, a researcher sampled 50 people
from a local gym who exercise regularly and 50 people from the surrounding
community who do not exercise regularly. Each subject reported to a clinic to have their cholesterol
measured. The subjects were unaware of the purpose of the study, and the technician measuring the
cholesterol was not aware of whether the subject exercises regularly or not. What type of study is
this?
A) An observational study.
B) An experiment, but not a double-blind experiment.
C) A double-blind experiment.
D )A matched-pairs experiment.
Answer: A
43. A correlation of zero between two quantitative variables means that
a) we have done something wrong in our calculation of the correlation
b) there is no association between the two variables
c) there is no linear association between the two variables
d) re-expressing the data will guarantee a linear association between the two variables Answer: C
44. Fill in the blank. A study is conducted to determine if one can predict the yield of a crop based on
the amount of yearly rainfall. The variable_______________ is the response variable in this study.
Answer yield of the crop
45. Fill in the blank. A researcher is interested in determining if one could predict the score on a
statistics exam from the amount of time spent studying for the exam. The variable_________is the
explanatory variable in this study. Answer: amount of time spent studying for the exam
46. Determine whether each of the following statements regarding the correlation coefficient is true
or false.
A) The correlation coefficient equals the proportion of times that two variables lie on a straight line.
B) The correlation coefficient will be +1.0 if all the data points lie on a perfectly horizontal straight
line.
C) The correlation coefficient measures the strength of any relationship that may be present between
two variables.
D) The correlation coefficient is a unitless number and must always lie between –1.0 and +1.0,
inclusive.
Answer: A) False, B) False, C) False, D) True
Use the following to answer questions 46 and 47: A group of college students believes that herbal
tea has remarkable restorative powers. To test its theory, the group makes weekly visits to a local
nursing home, visiting with residents, talking with them, and serving them herbal tea. After several
months, many of the residents are more cheerful and healthy.
47. What is the explanatory variable in this experiment?
A) The emotional state of the residents.
B) Herbal tea.
C) The fact that this is a local nursing home.
D) The college students. Answer: B
48. What is the lurking variable in this experiment?
A) The emotional state of the residents.
B) Herbal tea.
C) The fact that this is a local nursing home.
D) Visits of the college students.
Answer: D