BST 621 (Beasley) Homework 2 (100 points)
Use the following Sample Data for Questions 1 – 5.
Honda Toyota Nissan Mazda Mitsubishi
Male 60 58 84 46 32
Female 70 46 56 26 24
1. What is the Marginal Probability of randomly selecting a female from this sample of car owners?
P(F) = (2 points)
2. What is the Marginal Probability of randomly selecting a Honda owner from this sample of car owners? P(Honda) = (2 points)
3. What is the Conditional Probability of randomly selecting a Nissan owner given that she is female from this sample of car buyers? P(Nissan | F) = (2 points)
4. What is the Joint (Intersection) Probability of randomly selecting a Male Mazda owner from this sample of car buyers?
P(M ∩ Mazda) =
(2 points)
5. What is the Joint (Union) Probability of randomly selecting a Male or a Mitsubishi owner from this sample of car buyers? P(M U Mitsubishi) = (2 points)
6. Suppose flipping an unbiased coin four times. List all possible outcomes (show the entire sample space). (5 points)
7. What’s the Joint Probability of flipping 4 consecutive Heads? P(H = 4) =
(2 points)
8. What is the frequency distribution for the Sum of Number of Heads?
(3 points)
Number of Heads Frequency
0 Heads
1 Head
2 Heads
3 Heads
4 Heads
9. What is the Probability of flipping 3 or more Heads in 4 flips? P(H ≥ 3) =
(2 points)
1

BST 621 (Beasley) Homework 2 (100 points)
10. Suppose a garage door has a failure to open rate of P( X =1) = 0.05. That is, 5% of the time the garage door will not open when the remote is activated ( X =1). The other 95% of the time, the garage door opens ( X =0). Now suppose the garage door is to be opened N = 6 times in a day.
10.a.. List all possible combinations for r = 0 to 6 failures and using the binomial distribution function calculate the probability of each occurrence. (7 points)
N
C r
P[r] r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6
10.b. What is the probability of at least one failure, P( r ≥ 1) = _____________. (3 points)
10.c. Algebraically reduce the formula for the binomial distribution function and Write the general formula for the probability of at least one failure, (4 points)
P( r ≥ 1) =
11.
A microbiology laboratory records the number of “nuisance” bacteria on each specimen dish. Let x denote the number of “nuisance” bacteria on a randomly selected dish. The pdf of x is as follows:
(4 points)
Calculate the following probabilities directly from the pdf.
11.a. P( x = 2) =
11.b. P( x > 3) =
11.c. P(2 ≤
11.d. P(2 < x x
≤ 5) =
< 5) =


 

0
0.06
0.13
 0.20
x x x x




0
0
1
2
0.22
0.31
0.05
0.03
x x x x
11.e. Calculate the expected number of “nuisance” bacteria on each specimen dish,
(show your work) (3 points)




3
4
5
6
E(X) =
11.f. Calculate the variance of the number of “nuisance” bacteria on each specimen dish,
(show your work) (4 points)
Var(X) =
12 .
Suppose that only 20% of breast cancer patients complete online education. What is the probability that, of 20 randomly selected patients:
12.a. at most 6 will have completed online education?
12.b. exactly 6 will have completed online education?
(2 points)
(2 points)
12.c. at least 6 will have complete online education? (2 points)
12.d. How many of the next 20 patients do you expect to complete the online education? (4 points)
12.e. What is the variance associated with the expected value? (4 points)
Hint: The options are “complete” or “not-complete” so this is a trial with two possible outcomes.
You may use an on-line probability calculator or program this in Excel.
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BST 621 (Beasley) Homework 2 (100 points)
13. Take the data in BST611Assn2-2rolls.xls and Import it into SPSS, JMP, or SAS. 4. Use either
SPSS (Analyze-Descriptive Statistics-Explore); JMP (Analyze-Distribution); or SAS (PROC
UNIVARIATE; PROC BOXPLOT) to report the following statistics. (10 points)
Roll 1 Roll 2 Sum Mean
Y
1
= s
1
2
=
Y
2 s
2
2
=
=
Y
SUM
2 s
SUM
=
=
Y
Y s
Y
2
=
= s
1
= g
1
3
= g
1
4
= s
2 g
2
3
=
= s
SUM
3 g
SUM
=
= s
Y g
Y
3
=
= g
2
4
=
4 g
SUM
= g
Y
4
=
14. Using the concept of the Central Limit Theorem, describe how the Means and Variances for the roll of 2 dice (Mean and Sum) relate the Mean and Variance of a single roll of a die (Roll 1 or Roll 2).
(4 points)
15. Using the concept of the Central Limit Theorem, discuss the shape of the distribution of the Mean of Rolls as compared to the shape of the distribution of a Single Roll. (4 points)
16. Use either SPSS (Analyze-Correlate-Bivariate); JMP (Analyze-Multivariate Methods-
Multivariate); or SAS (PROC CORR) to compute the Pearson correlation between Rolls 1 and 2.
(2 points)
Pearson r
12
= ___________
17. What does this correlation ( r
12
) imply? (2 points)
18. Using the concept of the Central Limit Theorem, discuss the Mean, Variance, Standard Deviation, and the shape of the distribution of the Mean of Rolls for rolling N = 16 dice simultaneously.
(4 points)
19. If you were to roll N = 16 dice simultaneously, what is the Probability that the Mean Roll would be equal to or exceed 4.625 (sum of 74 or greater); P( Y ≥ 4.625) = (4 points)
20. Suppose the Distribution of GRE-Quantitative ( Q ) scores is Normal with a

= 500 with

= 100.
20.a. What is probability of randomly selecting a person with a score greater than or equal to 610?
P( Q ≥ 610) =
(4 points)
20.b. What is probability of randomly selecting N =10 people with a Mean score greater than or equal to 536? P( Q ≥ 536) =
(4 points)
3