SOLUTIONS TO THE THIRD MIDTERM EXAMINATION

SPRING, 2001

Chapter 6: Continuous Distributions

1. Times between accidents for all fatal accidents on scheduled American domestic passenger flights during the years 1948 through 1961 were found to have an exponential distribution with parameter q=44 days.

  1. Find the probability that the time between such accidents will be less than 30 days.

  2. Suppose that there were no such accidents for 30 days. What is the probability that there will not be any accidents for the next 44 days?

2. It is estimated that 80% of all 18-year-old women have weights ranging from 103.5 to 144.5 lb. Assuming the weight distribution can be adequately modeled by a normal distribution and assuming that 103.5 and 144.5 are equidistant from the average weight m, calculate s.

3. Let X denote the wing length in millimeters of a male gallinule and Y the wing length in millimeters of a female gallinule. Assume that X is normally distributed with mean 184.09 and standard deviation 6, and Y is normally distributed with mean 171.93 and standard deviation 7. Also, X and Y are independent.

  1. Find E[X-Y]=E[X]-E[Y]=184.09-171.93=12.16
  2. Find Var[X-Y]=Var[X]+Var[Y]=36+49=85
  3. What is the distribution of X-Y? N(12.16, 9.22)
  4. If a male and female gallinule are captured, what is the probability that X is greater than Y? (i.e. find P(X-Y>0)

Chapter 8 (Sampling Distribution of the Mean)

4. A random sample of size n=100 is to be taken from a uniform distribution with a=-1b=1.

  1. Based on the central limit theorem, what is the probability that the mean of the sample, , will exceed .10?

  2. By using the Chebyshev’s theorem find c and d such that
.

Chapter 8: Chi-Square, t, F distribution

5. Suppose that is a sample from a Normal distribution with m=2 s2=9 and .

  1. What is the distribution of ?

  2. What is the distribution of ?

  3. What is the distribution of
?

6.

  1. If and are the variances of independent random samples of size and from normal populations with , find

  2. Suppose that is the variance of a random sample of size 9 from a normal distribution with variance 4. Find
.

Order Statistics

7. Suppose that

  1. Find the sampling distribution of the largest order statistics Yn.

  2. Find the mean of Yn, i.e., E[Yn].

Note that: .