Is the Variable Number of Bananas in a Bunch Continuous or Discrete

(6 points) Classify each of the following random variables as discrete or continuous.
a) The weight of a bunch of bananas.
Continuous
b) The number of students in a statistics class.
Discrete

(11 points) Ford Motor Company's total sales revenues for 1986 are shown in the table below. Construct a relative frequency bar chart.

Region	Revenue ($billion)
U.S.	50
Canada	10
Europe	13
Latin Am.	3
All Other	2

Answer:

Relative Frequencies US	50/78 = 0.64 Can	10/78 = 0.13 Euro	13/78 = 0.17 L.A.	 3/78 = 0.04 Other	 2/78 = 0.03          

(7 points) Telephone calls put on hold have mean length 55 seconds and standard deviation 15 seconds. Find the z-score for a call that was put on hold for 82 seconds. If the distribution of holding times is mound-shaped, is an 82 second wait unusual? Why?

z = (x - )/ = (82-55)/15 = 27/15 = 1.8

This is 1.8 standard deviations above the mean. This is not unusual. 95% lie within 2 standard deviations.

(20 points) The salaries of a sample of 6 CEOs, in millions of dollars are:

2, 1, 5, 8, 5, 3.

Compute the following.

a) mean

= (2 + 1 + 5 + 8 + 5 + 3)/6 = 24/6 = 4

b) median

1, 2, 3, 5, 5, 8 median = (3 + 5)/2 = 4

c) mode

5

d) range

8 - 1 = 7

e) first quartile

i = 25/100 x 6 = 1.5 Round up to 2. Q1 = x₂ = 2.

f) third quartile

i = 75/100 x 6 = 4.5 Round up to 5. Q3 = x₅ = 5.

g) interquartile range

5 - 2 = 3

h) variance

s² = [ (2-4)² + (1-4)² + (5-4)² + (8-4)² + (5-4)² + (3-4)² ]/(6-1) = 32/5 = 6.4

i) standard deviation

(10 points) A boxplot is shown below for the number of farms in each state in the U.S. Estimate the values of the Five Number Summary from the boxplot.

            -----------------    ---I       |       I------------                    *       -----------------    +---------+---------+---------+---------+---------+-----           0        35        70       105       140       175

Minimum: 0
Q1: 10
Median: 37
Q3: 65
Maximum: 185

(9 points) A consultant has presented his client with three alternative research studies methods. If the client chooses alternative A, the consultant will have to hire 1 additional employee. Two employees will have to be hired if the client chooses B, and 5 if the client chooses C. The probability that the client will select alternative A is 0.1, with P(B) = 0.5 and P(C) = 0.4. Find the expected number of employees that will have to be hired.

            X         P(X)                X P(X)                 1         0.10                 0.10                  2         0.50                 1.00                  5         0.40                 2.00                                      E(X) =                        X P(X) =  3.10          

(9 points) A certain computer has two power supplies. The computer can operate on either one. On any given day, the first is 99% reliable and the second is 99.5% reliable. What is the probability that on any given day the computer will lose its power?

P(lose power) = P(first fails and second fails)
= P(first fails second fails)
= P(first fails) P(second fails)
= (0.01) (0.005)
= 0.00005

(9 points) The results of an experiment studying children's understanding of TV commercials are shown in the table.

            Age                                        5-7    8-10      11-13    Don't understand        55     65         30      Understand              45     35         70          

1. What is the probability that a 9 year old will understand TV commercials?

   P(understanding | 9 years old) = 35/(65 + 35) = 0.35

2. Is understanding independent of age? Use probabilities to demonstrate your answer.

   P(understanding) = (45+35+70)/300 = 0.50

   P(understanding | 8-10 years old) = 35/(65+35) = 0.35

   These are not equal, so understanding and age are dependent.

(15 points) A production line produces bolts. Machine X produces 60% of the bolts and Machine Y produces the remainder. 0.1% of the bolts produced on Machine X are defective and 0.2% of the bolts produced on Machine Y are defective.

1. Construct a probability tree for this situation.

   

2. What is the probability of a bolt being defective?

   P(D) = P(X and D or Y and D) = P(X D) + P(Y D)
   = 0.0006 + 0.0008 = 0.0014

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Source: http://www.ma.iup.edu/~hedonley/ma214/t1s95/