Question

Edith Educationer has a real problem with children’s television programs.  She believes there are too many commercials in them.  To study this, she collects a sample of 500 children’s programs (n = 500) and counts how many commercials are on during the program.  The frequency count of programs with the number of commercials is found below:  

Commercials, X	8	9	10	11	12
Frequency	50	75	150	125	100
P(X)

Complete the table to produce the general discrete probability distribution (3.75 points).  Use this probability distribution to answer questions a – h.  

1. How many commercials, X, is the most likely outcome to occur?  
2. Find the average number and standard deviation of commercials a child will expect to see in a program.
3. What is the probability that a child will see 10 or more commercials when watching a program?  P(X ≥ 10)
4. What is the probability that a child will watch 4 commercials?  P(X <4)
5. What is the probability that a child will watch exactly 12 commercials?  P(X =12)
6. What is the probability that a child will watch exactly 7 commercials?  P(X =7)
7. What is the expected number of commercials in a program
8. Graph the probability Distribution

Answer 1

X              P(X)

8               50/500 = 0.1

9               75/500 = 0.15

10             150/500 = 0.3

11             125/500 = 0.25

12             100/500 = 0.2

a) X is most likely 10 to occur. Because it has highest probability.

b) Average number = E(X) = 8 * 0.1 + 9 * 0.15 + 10 * 0.3 + 11 * 0.25 + 12 * 0.2 = 10.3

E(X²) = 8² * 0.1 + 9² * 0.15 + 10² * 0.3 + 11² * 0.25 + 12² * 0.2 = 107.6

Var(X) = E(X²) - (E(X))² = 107.6 - 10.3² = 1.51

Standard deviation = sqrt(1.51) = 1.23

c) P(X > 10) = P(X = 10) + P(X = 11) + P(X = 12) = 0.3 + 0.25 + 0.2 = 0.75

d) P(X < 4) = 0

e) P(X = 12) = 0.2

f) P(X = 7) = 0

g) Expected number = average number = 10.3 or 10 (approx)

h)