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Stat 578 Assignment 2 Solutions
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Assignments2 solutions: Due by Midnight Sunday June 16th, 2013
(drop box of week 2) (Chapters 4, 5, and 6) Total 125 points.

True/False (2 points each)

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?

  1. 0.06
  2. 0.26
  3. 0.23
  4. 0.15
  5. 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

             

 

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