For each of the following two parts, you must justify your answer

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1.  True or False

For each of the following two parts, you must justify your answer.  No explanation = no credit.  Partially correct explanations=partial credit.

a.)    If a random variable X has a uniform distribution between the values 1 and 5, then the value of the cumulative distribution function at X=4 is .75.

b.)    For a confidence interval, all else equal, if we increase the number of observations by a factor of 4 (that is, collect 4x as much data) then the width of the interval will be reduced by 1/4.

2.A continuous random variable X has a uniform distribution over the interval (-k, 3k), where k is some number.  The height of the PDF over this interval is .05. For all other values of X, the PDF=0.

a.) What is the value of k?
b.) What is the probability that X=7.5?
c.) What is the probability that a randomly selected X lies between -10 and 10?

3.In each of 7 randomly selected hours, the following number of cars pass through a toll booth exit on the PA Turnpike:
 

Cars

8

10

7

10

5

12

18


a.    (10 points) What is the sample mean  ?
b.    (10 points) What is the sample variance  ?
The sample variance is,
c.    (10 points) What is the 99% confidence interval for μx, the population average number of cars in an hour. 

4.The following problems use the following joint probability table.  
 

 

Y = 1

Y = 7

 Marginal Probability

X = 3

0.15

0.4

0.55

X = 5

0.2

0.25

0.45

 Marginal Probability

0.35

0.65

1


a) What are E[X], E[Y]?
b)  What is the conditional expectation of Y when X=5?
c) What is the covariance between X and Y?
d) You could (but shouldn’t) show that the variance of X is .99 and the variance of Y is 8.19.  What is the expected value and variance of the following term: 7+8X-5Y?

5.Suppose a selective university only considers accepting students with a cumulative grade point average of 3.15 or higher.  Students below this threshold are not considered. Suppose the population of students applying to this university has a GPA that is normally distributed with a mean of 3.09 and standard deviation .4.  

a) What is the probability that a randomly selected student will meet the GPA threshold?

b) If 50 applicants are chosen at random, what is the probability that 20 or more of them will meet the GPA threshold? (Note: You’ll want to use your answer from part a.  If you’re not sure about this answer (and even if you are), be very clear about your calculations for part b.)

6.Suppose 60 stock market mutual funds are chosen at random and their results for the most recent year are recorded.  Of these, after fees and expenses, 13 of them earned their investors a return higher than the stock market as a whole. (Note for those interested in investing: this is a pretty realistic figure.  Most mutual funds fail to overperform the broader market in a year.)

What is the 85% confidence interval for the population proportion of mutual funds that have a return higher than the stock market as a whole?

 

  • 9 years ago
For each of the following two parts, you must justify your answer
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