Question

The data and scenario from this question follows into the next question. Assume that for an ad campaign to be successful, at least 76% of those seeing a television commercial must be able to recall the name of the company featured in the commercial one hour after viewing the commercial. Before distributing an ad campaign nationally, an advertising company plans to show the commercial to a random sample of 24 people. It will also show the same people two additional commercials for different products or businesses. a. The expected number of people is 18.24 (Type an integer or a decimal. Do not round.) b. Suppose that in the sample of 24 people, 14 were able to recall the name of the company in the commercial one hour after viewing. Based on the premise that the advertising campaign will be successful, what is the probability of 14 or fewer people being able to recall the company name? The probability is (Round to four decimal places as needed.)

Answer 1

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is executed
p = success probability
mean = 24 * 0.76
= 18.24
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 24 * 0.76 * 0.24
= 4.3776
III.
standard deviation = sqrt( variance ) = sqrt(4.3776)
=2.0923
a.
Expected number of people is 18.24
b.
probability that of 14 or fewer people being able to recall company name
P( X < = 14) = P(X=14) + P(X=13) + P(X=12) + P(X=11) + P(X=10) + P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4)   
= ( 24 14 ) * 0.76^14 * ( 1- 0.76 ) ^10 + ( 24 13 ) * 0.76^13 * ( 1- 0.76 ) ^11 + ( 24 12 ) * 0.76^12 * ( 1- 0.76 ) ^12 + ( 24 11 ) * 0.76^11 * ( 1- 0.76 ) ^13 + ( 24 10 ) * 0.76^10 * ( 1- 0.76 ) ^14 + ( 24 9 ) * 0.76^9 * ( 1- 0.76 ) ^15 + ( 24 8 ) * 0.76^8 * ( 1- 0.76 ) ^16 + ( 24 7 ) * 0.76^7 * ( 1- 0.76 ) ^17 + ( 24 6 ) * 0.76^6 * ( 1- 0.76 ) ^18 + ( 24 5 ) * 0.76^5 * ( 1- 0.76 ) ^19 + ( 24 4 ) * 0.76^4 * ( 1- 0.76 ) ^20
= 0.0425