a. (4 pts) Let X be the number of calls that are from commercial businesses.

4. Calls coming into a customer service center for a particular internet company are from commercial businesses 30% of the time and private households 70% of the time, and calls are independent from each other.  The manager observes a total of ten calls.

Minitab: Calc > Probability Distributions > Binomial. 

Minitab Express: Statistics > Probability Distributions > CDF/PDF > Cumulative (CDF) or Probability (PDF).

a.      (4 pts) Let X be the number of calls that are from commercial businesses.  Then, X is a binomial random variable.  What are the parameters of X?

              n =

              p =

b.      (4 pts) Use software (or by hand) to find the probability that out of the 10 calls, exactly one will be from a commercial business.

c.       (4 pts) Use software (or by hand) to find the probability that no more than one call is from a commercial business.

d.      (4 pts) By hand, find the probability that at least two calls are from a commercial business.  To do this, use the probability relationship of complements, P(A)  = 1 – P(A^C), and your answer from part c.

e.       (6 pts) Use software and basic arithmeticto find the probability that more than 2, but no more than 6, calls are from commercial businesses.  Be sure to paste any relevant output below.

      Hint: P(2 < X ≤ 6) = P(X ≤ 6) – P(X ≤ 2)

f.        (6 pts)  By hand, find the expected value and standard deviation of X using the parameters found in part a. Show all work.

Expected value =

             Standard deviation =

5.  For those intending to major in Business, quantitative GRE scores are normally distributed with mean 152 and standard deviation 8.  Let Y denote a randomly selected quantitative GRE score.  Use this to answer the following questions.

Minitab: Calc > Probability Distributions > Normal. 

Minitab Express: Statistics > Probability Distributions > CDF/PDF > Cumulative (CDF).

a.      (4 pts) Use software to find P(Y ≤ 150).  Be sure to paste your output.

b.      (4 pts) Write a sentence interpreting the value you found above, in context of the scenario.  You should write the sentence so that it makes sense to someone not taking a statistics course.

c.       (4 pts) A certain university will only consider admitting applicants who have a quantitative GRE of at least 150.  Use your answer from part a to find the probability that a randomly selected Business applicant will be considered for admittance.

d.      (6 pts) Use software to find the probability that an intended Business major scores between 140 and 160 on the quantitative GRE.

Hint: P(140 < Y ≤ 160) = P(Y ≤ 160) – P(Y ≤ 140)

e.       Let’s redo part a using the standard normal probability table.  To do this, follow the steps below.

                                i.            (4 pts) Calculate the z-score of Y = 150. Round to two decimal places.

                              ii.            (2 pts) Use the standard normal probability table and your answer to part i to find P(Y ≤ 150).  Show all 4 decimal places as given in the table.

f.        (6 pts) A prestigious university will only considering admitting Business students who scored in at least the 80^th percentile of the quantitative GREs (in other words, the student must have scored higher than 80% of his peers).  What is the minimum score would you need to get in order to be considered for admittance into the university?  You may use software or the normal probability table to find the answer, but please show your work.

Hint: This is also called an inverse probability, because instead of finding a probability using a z-score/value, you find the z-score/value based on a cumulative probability (aka percentile).  See the bottom of Section 5.6 of the online notes.  In this case, you want to find a value y such that P(Y ≤ y) = .80.