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# Coefficient of variation, probability, normal distribution

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1) The spread in the annual prices of stocks selling for under \$10 and the spread in prices of those selling for over \$60 are to be compared. The mean price of the stocks selling for under \$10.00 is \$5.25 and the standard deviation \$1.52. The mean price of those stocks selling for over \$60 is \$92.50 and the standard deviation \$5.28.

a. Why should the coefficient of variation be used to compare the dispersion in the prices?

b. Compute the coefficients of variation. What is your conclusion?

2) A study of 200 grocery chains revealed these incomes after taxes:

Income after Taxes Number of Firms

Under \$1 million 102
\$1 million to \$20 million 61
\$20 million or more 37

a. What is the probability a particular chain has under \$1 million in income after taxes?
b. What is the probability a grocery chain selected at random has either an income between \$1 million and \$20 million, or an income of \$20 million or more? What rule of probability was applied?

3) A survey of undergraduate students in the school of business at Northern University revealed the following regarding the gender and majors of the students:

Major

Gender Accounting Management Finance Total

Male 100 150 50 300
Female 100 50 50 200
Total 200 200 100 500

a. What is the probability of selecting a female student?
b. What is the probability of selecting a finance or accounting major?
c. What is the probability of selecting a female or an accounting major? Which rule of addition did you apply?
d. Are gender and major independent? Why?
e. What is the probability of selecting an accounting major given that the person selected is a male?
f. Suppose two students are selected randomly to attend a lunch with the president of the university. What is the probability that both of those selected are accounting majors?

4) The accounting department at Weston Materials, Inc, a national manufacturer or unattached garages reports that it takes two construction workers a mean of 32 hours and a standard deviation of 2 hours to erect the Red Barn model. Assume the assembly times follow the normal distribution.

a. Determine the z values for 29 and 34 hours. What percent of the garages take between 32 hours and 34 hours to erect?
b. What percent of the garages take between 29 hours and 34 hours to erect?
c. What percent of the garages take 28.7 hours or less to erect?
d. Of the garages, 5 percent take how many hours or more to erect?

5) The number of passengers on the Carnival Sensation during one-week cruises in the Caribbean follows the normal distribution. The mean number of passengers per cruise is 1,820 and the standard deviation is 120.

a. What percent of the cruises will have between 1820 passengers or more?
b. What percent of the cruises will have 1970 passengers or more?
c. What percent of the cruises will have 1600 or fewer passengers?
d. How many passengers are on the cruises with the fewest 25 percent of passengers?

6) In establishing warranties on HDTV sets, the manufacture wants to set the limits so that few will need repair at manufacturer expense. On the other hand, the warranty period must be long enough to make the purchase attractive to the buyer. For a new HDTV the mean number of months until repairs are needed is 36.84 with a standard deviation of 3.34 months. Where should the warranty limits be set so that only 10 percent of the HDTV's need repairs at the manufacturer's expense?

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#### Solution Preview

The spread in the annual prices of stocks selling for under \$10 and the spread in prices of those selling for over \$60 are to be compared. The mean price of the stocks selling for under \$10.00 is \$5.25 and the standard deviation \$1.52. The mean price of those stocks selling for over \$60 is \$92.50 and the standard deviation \$5.28.

a. Why should the coefficient of variation be used to compare the dispersion in the prices?

Coefficient of variation = Standard deviation / Mean
Coefficient of variation of variation is used since a spread of \$1 for a \$5 stock = 1/5 =20% of the price whereas a spread of \$ 1 for a \$ 50 stock is only 2% of the price. Thus a standard deviation of \$1 is not the same for both stocks.

b. Compute the coefficients of variation. What is your conclusion?

Under \$ 10 stock:
Mean= \$5.25
Standard deviation= \$1.52
Therefore, Coefficient of variation= Standard deviation / Mean= 0.2895 or 28.95% =1.52/5.25

Over \$ 60 stock:
Mean= \$92.50
Standard deviation= \$5.28
Therefore, Coefficient of variation= Standard deviation / Mean= 0.0571 or 5.71% =5.28/92.5

Thus, under \$10 stocks are riskier as they have a higher coefficient of variation

A study of 200 grocery chains revealed these incomes after taxes:

Income after Taxes Number of Firms
Under \$1 million 102
\$1 million to \$20 million 61
\$20 million or more 37
Total 200

a. What is the probability a particular chain has under \$1 million in income after taxes?

# of firms with income under \$ 1 million= 102
# of firms = 200
Probability of firms under \$ 1million = 0.51 =102/200

b. What is the probability a grocery chain selected at random has either an income between \$1 million and \$20 million, or an income of \$20 million or more? What rule of probability was applied?

# of firms with income between \$ 1 million and \$ 20 million= 61
# of firms = 200
Probability of firms between \$1 -\$20 million = 0.305 =61/200

# of firms with income greater than \$ 20 million= 37
# of firms = 200
Probability of firms greater than \$20 million = 0.185 =37/200

Therefore, probability a grocery chain selected at random has either an income between \$1 million and \$20 million, or an income of \$20 million or more=
= 0.49 =0.305+0.185

The probability rule applied is addition rule of probability

A survey of undergraduate students in the school of business at Northern University revealed the following regarding the gender and majors of the students:

Major

...

#### Solution Summary

This solution answers questions on coefficient of variation, probability, and normal distribution using simple calculations.

\$2.49