# Determining range, mean absolute deviation, standard deviation and variance

1. A social scientist for a children's advocacy organization has randomly selected 10 Saturday-mornings television cartoon shows and carried out a content analysis in which he counts the number of incidents of verbal or physical violence in each. For 10 cartoons examined the counts were as follows: 27, 12, 16, 22, 15, 30, 14, 30, 11, and 21. Determine the mean and the median for these data. Is there a mode? If so, what is its value?

2. For the sample of Saturday-morning cartoon violence counts described in problem number 1. Determine the range, the mean absolute deviation, the standard deviation, and the variance.

3. The manufacturer of an extended-life light-bulb claims the bulb has an average life of 12,000 hours, with a standard deviation of 500 hours. If the distribution is a bell shaped and symmetrical, what is the approximate percentage of these bulbs that will last

a. Between 11,000 and 13,000 hours?

b. Over 12,500 hours?

c. Less than 11,000 hours?

d. Between 11,500 and 13,000 hours?

4. Explain what is meant by sampling error, response error, and non-response error in survey research.

Chapter 4 and 5

1. If a die is rolled one time , classical probability would indicate that the probability of a "two" should be 1/6, If the die rolled 60 times and comes up "two" only 9 times , does this suggest that the die is "loaded"? Why or why not?

2. It has been reported that about 35% of U.S. adults attend a sports event during previous years. What are the odds that a randomly selected U.S. adult attended a sports event during the year?

3. The following contingency table of frequencies is based on a 5-year study of fire fatalities in Maryland. For purposes of charity, columns and rows are identified by letters A-C and D-G respectively.

Age A 0.00% B 0.01-0.09% C ≥0.10%

D 0-19 142 7 6 155

E 20-39 47 8 41 96

F 40-59 29 8 77 114

G 60 and over 47 7 35 89

265 30 159 454

a. For this table, identify any two events that are mutually exclusive.

b. For this table, identify any two events that intersect.

4. Using the information presented in the table from problem 3, calculate the following probabilities:

a. P(A or D)

b. P(B or F)

c. P( C or G)

d. P (B or C or G)

5. A fair coin tossed three times. What is the probability that the sequence will be heads, tails, and heads?

6. Using the table in problem 3, calculate the conditional probability of C given each of the age groups, or P (CD), P (CE), etc. Compare these probabilities and speculate as to which age groups seem more likely than others to have been ( according to the legal definition at that time, 0.10% blood alcohol content) intoxicated at the time they were victims.

7. An investment counselor would like to meet with 12 of his clients on Monday, but he has time for only 8 appointments. How many different combinations of the clients could be considered for inclusion into his limited schedule for that day?

8. A roadside museum has 25 exhibits but enough space to display only 10 at time. If the order of arrangement is considered, how many possibilities exist for the eventual display?

#### Solution Preview

Mean = ( 27 + 12 + 16 + 22 + 15+ 30 + 14 + 30 + 11 + 21 ) / 10

= 19.8

Median:

Arranging the data in the ascending order

11 12 14 15 16 21 22 27 30 30

So the median = ( 16 + 21) /2

= 18.5

Mode:

The value 30 has repeated twice , so Mode is 30.

2. For the sample of Saturday-morning cartoon violence counts described in problem number 1. Determine the range, the mean absolute deviation, the standard deviation, and the variance.

Solution:

Range = 30 - 11 = 19.

E[X2] = ( 112 + 122 + 142 + 152 + 162 + 212 + 222 + 272 + 302 + 302)

= 439.6

Var[X] = 439.6 - 19.82

= 47.56

SD = Sqrt(Var[X])

= 6.896

Cartoons Deviation from the Mean Absolute Deviation from the mean

11 -8.8 8.8

12 -7.8 7.8

14 -5.8 5.8

15 -4.8 4.8

16 -3.8 3.8

21 1.2 1.2

22 2.2 2.2

27 5.2 5.2

30 8.2 8.2

30 8.2 8.2

Total 56

MAD (Mean Absolute Deviation) is (∑|xi - mean|)/ 10 = 56 /10 = 5.6.

3. The manufacturer of an extended-life light-bulb claims the bulb has an average life of 12,000 hours, with a standard deviation of 500 hours. If the distribution is a bell shaped and symmetrical, what is the approximate percentage of these bulbs that will last

Solution:

a. Between 11,000 and 13,000 hours?

P( 11000<X<13000 ) = ...

#### Solution Summary

The solution determines the range, mean absolute deviation, standard deviation and the variance of a data set. The mean, median and mode are determined and explained.