Formulas:

	E(x) = S x p(x)
V(x) = S x2 p(x) - (S x p(x))2

1. (10 points) Find the expected number of breakdowns in a week for a machine if the probability of no breakdowns in a week is 0.30, the probability of one breakdown is 0.60, and the probability of two breakdowns is 0.10.

   # of breakdowns (x)	probability (p(x))	x p(x)
   0	0.30	0 (0.30) = 0
   1	0.60	1 (0.60) = 0.60
   2	0.10	2 (0.10) = 0.20
   	Total	E(x) = S x p(x) = 0.80

2. (10 points) Suppose that 20% of all flights at a particular airport experience delays. If an executive takes 10 flights from the airport next month, find the probability that at least 5 of the flights will be delayed.

   Binomial
   n = 10, p = 0.20, q = 0.80

   Find P(X > 5) = 1 - P(X < 4) = 1 - 0.967 (from binomial table) = 0.033

3. (10 points) Suppose that a production facility experiences an average of 3 accidents per week. Find the probability that in a particular week exactly 5 accidents will occur.

   Poisson

   m = 3

   Find P(X = 3) =

   or using the Poisson table,

   P(X = 5) = P(X < 5) - P(X < 4) = 0.916 - 0.815 = 0.101

4. (10 points) The research and development section of a particular company must submit proposals to upper management before embarking on any large research projects. Suppose that 25% of these proposals are accepted. Find the probability that among 11 recently submitted proposals, exactly 4 of them will be accepted.

   Binomial
   n = 11, p = 0.25, q = 0.75

   

   
   

5. (10 points) Suppose that a hospital's pharmaceutical delivery every morning is equally likely to occur at any time between 7:00 a.m. and 9:00 a.m. Find the probability that tomorrow's delivery will occur between 7:15 a.m. and 7:30 a.m.

   

   Uniform

   7:15 -> 7.25 hours

   7:30 -> 7.5 hours

   P(7.25 < X < 7.5) = (7.5 - 7.25) = 0.25/2 = 0.125

   
   

6. (8 points) Let z be a standard normal random variable. Find P(z > -1.35).

   

   P(z > -1.35) = 0.5 + P(0 < z < 1.35) = 0.5 + 0.4115 = 0.9115

   
   

7. (8 points) Let x be a normal random variable with u = 4.5 and s = 2. Find P(3 < x < 5).

   

   Convert to z

   

   

   
   

   P(3 < x < 5) = P(-0.75 < z < 0.25) = 0.2734 + 0.0987 = 0.3721

   
   

8. (8 points) Let z be a standard normal random variable. Find a so P(z > a) = 0.2.

   

   Search for 0.3000 in the interior of the normal table.

   a = 0.84.

9. (10 points) Suppose that 90% of a manufacturer's computer chips conform to specifications. What is the probability that in a sample of 120 chips that between 80% and 85% conform to the specifications?

   Binomial

   n = 120, p = 0.90, q = 0.10

   Find P(0.80 < < 0.85)

   Check for normality:

   np = (120)(0.90) = 108 > 5
   nq = (120)(0.10) = 12 > 5

   

   Convert to z

   

   

   

   P(0.80 < < 0.85) = P(-3.65 < z < -1.83) = 0.5 - 0.4664 = 0.0336

10. (10 points) Suppose that cans of salmon have an average weight of 6.05 ounces with standard deviation 0.18 ounces. Find the probability that a random sample of 36 cans has an average weight of more than 6 ounces.

    n = 36
    m = 6.05
    s = 0.18

    Find P( > 6)

    Normal, since n > 30
    
    

    Convert to z

    

    P( > 6) = P(z > -1.67) = 0.5 + 0.4525 = 0.9525