Chapter 4
1. If events A and B are independent and A is not an impossible event, then P(A/B) is not equal to zero. TRUE In fact P(A/B) equals P(A) if A and B are independent, which is not zero unless A is an impossible event.
2. If events A and B are mutually exclusive, then P(A/B) is equal to zero. TRUE This is obvious from the definition of mutually exclusive events. If B occurs then A cannot occur at the same time. Therefore P(A/B) = 0.
3. The union of events A and B is given by all basic outcomes common to both A and B FALSE
Chapter 5
4. If the probability of success is 0.4 and the number of trials in a binomial distribution is 150, then its standard deviation is 36. FALSE σ= √(np(1-p)) =√(150*0.4*0.6) = 6
5. If a fair coin is tossed 100 times, then the variance of the random variable defined as the number of heads is exactly five. FALSE σ2 = np(1-p)= 100*0.5*0.5= 25. So the standard dev is 5 not the Variance.
6. If a fair coin is tossed 20 times then the probability of exactly 10 Tails is more than 18 percent.
FALSE It is 17.62 percent
7. The probability that a person catches a cold during the cold and flu season is 0.4. If 10 people are chosen at random, the standard deviation for the number of persons catching cold is 1.55. (Hint: convert the problem to a binomial distribution problem). TRUE Here p = 0.4 and n=10. Therefore, standard deviation = sq root of 10*0.4*0.6
Chapter 6
8. The number of defective pencils in a lot of 1000 is an example of a continuous random variable. FALSE It is a result of counting- so discrete.
9. For a continuous distribution, P(X ≤ 100) = P(X < 100). TRUE See my Instructions on this.
10. All continuous random variables are normally distributed.
FALSE Continuous random variables can be highly skewed and non-normal. Even if it is symmetrical it may not be normal but other distribution like t-distribution. A normal random variable is a popular example of a continuous random variable, but a continuous r.v. need not be normal.
11. The mean of a standard normal distribution is always equal to 1. FALSE. Its mean is zero and variance (or std deviation) equal to 1.
12. If the sample size is as large as 1000, we can safely use the normal approximation to binomial even for small p. FALSE (Instructions on Ch6) : For example if p is .001 then np would be only 1 even if sample size is 1000.
Multiple Choice (3 points each)
Chapter 4
1. Two mutually exclusive events having positive probabilities are ______________ dependent.
A. Never
B. Sometimes
C. Always
They are necessarily dependent because the occurrence of one (seriously) affects the probability of the other (makes it zero). Instructions on Ch 4 page 4
2. If P(A) >0 and P(B) > 0 and events A and B are independent, then:
A. P(A) = P(B)
B. P((A|B)) = P(A)
C. P(A B) = 0
D. P(A B)=P(A)/ P(B/A)
E. Both A and C are correct
See My Instructions on Ch 4 page 5. Independence does not imply equality of probabilities. So the first choice is clearly wrong. The third choice applies to mutually exclusive events not independent events. The fourth choice is also incorrect because there should be multiplication on the right hand side not division. So the correct answer is B.
3. A recent marketing survey tried to relate a consumer's awareness of a new marketing campaign with their rating of the product. Consumers rated their awareness as low, medium and high, and rated the product as poor, fair, or good. The results are presented below:
Rating Awareness
Low Medium High
Poor 0.10 0.15 0.07
Fair 0.06 0.11 0.06
Good 0.07 0.11 0.27
What is the probability that a consumer who ranked the product as fair had a high awareness of the ad campaign?
A. 0.06
B. 0.26
C. 0.23
D. 0.15
E. 0.40
The completed table is:
Rating Awareness
Low Medium High
total
Poor 0.10 0.15 0.07 0.32
Fair 0.06 0.11 0.06 0.23
Good 0.07 0.11 0.27 0.45
Total 0.23 0.37 0.40 1.00
P(High Awareness/Fair) = 0.06/0.23 = 0.26 rounded to two decimal places.
4. With the data in question 3 above, what is the probability that a randomly selected consumer either has a low awareness or rated the product poor?
A. 0.23
B. 0.32
C. 0.55
D. 0.45
E. 0.10
The completed table is:
Rating Awareness
Low Medium High
total
Poor 0.10 0.15 0.07 0.32
Fair 0.06 0.11 0.06 0.23
Good 0.07 0.11 0.27 0.45
Total 0.23 0.37 0.40 1.00
P( Low awareness or Poor rating) = 0.23 + 0.32 - 0.10 = 0.45
Chapter 5
5. In a study conducted by UCLA, it was found that 25% of college freshmen support increased military spending. If 6 college freshmen are randomly selected, find the probability that: at least 3 support increased military spending
A. 0.8306
B. 0.1318
C. 0.1694
D.0 .9624
Remember that at least 3 means 3 or more.(You can use Table on page 855 for n=6 and p=.25) or use computer to get the following: Binomial distribution : n= 6 p= 0.25
cumulative
X P(X) probability
0 0.17798 0.17798
1 0.35596 0.53394
2 0.29663 0.83057
3 0.13184 0.96240
4 0.03296 0.99536
5 0.00439 0.99976
6 0.00024 1.00000
1.00000
6. A fair die is rolled 10 times. What is the probability that an even number (2, 4 or 6) will occur more than 3 times?
A. 0.1719
B. 0.1172
C. 0.6230
D. 0.9453
E. 0.8281
Here n =10 and p = 0.5. We need P(X > 3) which is 1- P(X ≤ 3) = 1- [P(X= 0) + P(X = 1) + P(X = 2) + P(X=3)] = 1- 0.1719 = 0.8281. From the Table on page 854 you get the answer. Or you can use computer as follows:
Binomial distribution
10 n
0.5 p
cumulative
X P(X) probability
0 0.00098 0.00098
1 0.00977 0.01074
2 0.04395 0.05469
3 0.11719 0.17188
4 0.20508 0.37695
5 0.24609 0.62305
6 0.20508 0.82813
7 0.11719 0.94531
8 0.04395 0.98926
9 0.00977 0.99902
10 0.00098 1.00000
1.00000
7. Which one of the following statements is not a necessary assumption of the binomial distribution?
A. Sampling is with replacement
B. The experiment consists of n identical trials
C. The probability of success remains constant from trial to trial
D. Trials are independent of each other
E. Each trial results in one of two mutually exclusive outcomes
8. If p = 0.6 and n =10, then the corresponding binomial distribution is
A. Right skewed
B. Left skewed
C. Symmetric
D. Bimodal
If p = 0.5 it is symmetric for any number of trials. If p is larger than 0.5, then the distribution is left skewed. If p is smaller than 0.5, then it is right skewed.
9. A fair die is rolled 64 times. What is the standard deviation of the even number (2, 4 or 6) outcomes?
A. 32
B. 16
C. 4
D. 8
The probability of 2 or 4 or 6 in any trial is 0.5 and there are 64 trials. Therefore, σ = √(64*0.5*0.5) = 4 using the formula for binomial distribution.
10. A fair die is rolled 10 times. What is the probability that an odd number (1, 3 or 5) will occur less than 4 times?
A. 0.1719
B. 0.1172
C. 0.2051
D. 0.3770
E. 0.0439
Add P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) from table on page 856 or can use excel to find the cumulative probability
11. In the most recent election, 20% of all eligible college students voted. If a random sample of 20 students were surveyed:
Find the probability that at least five students voted in the election.
A. 0.1746
B. 0.2182
C. 0.3704
D.0.6296
P(X ≥ 5)= 1- P(X ? 4) = 1- 0.629648 (using excel's Binomdist) or using table and adding the probabilities up to 4 and subtracting from 1.
Chapter 6
12. The area under the normal curve between z = 0 and z = 1 is ________________ the area under the normal curve between z =1 and z = 2.
A. Greater than
B. Less than
C. Equal to
D. A, B or C above dependent on the value of the mean
E. A, B or C above dependent on the value of the standard deviation
Remember the height of the curve declines as we move farther from the mean value 0. Therefore, the area under the curve (or probability will decrease for the same interval length as we move farther from the mean.
13. The price-to-earnings ratio for firms in a given industry is distributed according to normal distribution. In this industry, a firm with a Z score for its price-to-earnings ratio equal to 0.5:
A. Has an average price-to-earnings ratio
B. Has a below average price-to-earnings ratio
C. Has an above average price-to-earnings ratio
D. May have an above average or below average price-to-earnings ratio
The mean or average for Z is 0. Therefore the given value is above average.
14. The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent more than 90 days overdue. The historical records of the company show that over the past 8 years 14 percent of the accounts are delinquent. For this quarter, the auditing staff randomly selected 250 customer accounts. What is the probability that less than 30 accounts will be classified as delinquent?
A. 31.86%
B. 18.14%
C. 81.86%
D. 63.72%
E. 75.84%
Here nπ = 250*(0.14) = 35, n(1-π) = 215 and nπ(1-π) = 30.1 all ≥ 10. So, normal approximation without continuity correction can be performed. You can work with X or proportion to get the same answer. Let us work with X first. Mean nπ = 35. Variance = nπ(1-π) = 30. Standard deviation (or standard error) = √30 = 5.4863.
The Z score of 30 = (30-35)/5.4863 = -0.91 (rounded to two decimal places).
Now I will show you that the same result is obtained working with proportions. The mean of the sample proportion (or its expected value) is 0.14 (the population proportion). The std error is √{π(1-π)/n} = 0.0219 and given value p = 30/250 = 0.12
Therefore Z = (0.12 - 0.14)/0.0219 = - 0.91 (rounded to two decimal places).
P(X 30) = P(Z -0.91) = 0.1814 from table or MegaStat
Normal distribution
P(lower) P(upper) z
0.8186 0.1814 0.91
0.1814 0.8186 -0.91