A hotel claims that 85 % of its customers are very satisfied with its service. C

Question

A hotel claims that 85 % of its customers are very satisfied with its service. Complete parts a through d below based on a random sample of eight customers.

a. What is the probability that exactly seven customers are very satisfied?

_______ (Round to four decimal places as needed.)

b. What is the probability that more than seven customers are very satisfied?

______ (Round to four decimal places as needed.)

c. What is the probability that less than six customers are very satisfied?

_______ (Round to four decimal places as needed.)

d. Suppose that of eight customers selected, two responded that they are very satisfied. What conclusions can be drawn about the hotel's claim?

The probability that 2 out of 8 customers are very satisfied is _______ assuming that 85 % of customers are very satisfied.
Based on this sample, it is (likely or not likely) ? that 85 % of customers are very satisfied. (Round to four decimal places as needed.)

Explanation / Answer

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 excueted
p = success probability
mean = 8 * 0.85
= 6.8
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 8 * 0.85 * 0.15
= 1.02
III.
standard deviation = sqrt( variance ) = sqrt(1.02
=1.00995
a.
P( X = 7 ) = ( 8 7 ) * ( 0.85^7) * ( 1 - 0.85 )^1
= 0.38469
b.
P( X < = 7) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 8 7 ) * 0.85^7 * ( 1- 0.85 ) ^1 + ( 8 6 ) * 0.85^6 * ( 1- 0.85 ) ^2 + ( 8 5 ) * 0.85^5 * ( 1- 0.85 ) ^3 + ( 8 4 ) * 0.85^4 * ( 1- 0.85 ) ^4 + ( 8 3 ) * 0.85^3 * ( 1- 0.85 ) ^5 + ( 8 2 ) * 0.85^2 * ( 1- 0.85 ) ^6 + ( 8 1 ) * 0.85^1 * ( 1- 0.85 ) ^7 + ( 8 0 ) * 0.85^0 * ( 1- 0.85 ) ^8   
= 0.72751
P( X > 7) = 1 - P ( X <=7) = 1 -0.72751 = 0.27249
c.
P( X < 6) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 8 5 ) * 0.85^5 * ( 1- 0.85 ) ^3 + ( 8 4 ) * 0.85^4 * ( 1- 0.85 ) ^4 + ( 8 3 ) * 0.85^3 * ( 1- 0.85 ) ^5 + ( 8 2 ) * 0.85^2 * ( 1- 0.85 ) ^6 + ( 8 1 ) * 0.85^1 * ( 1- 0.85 ) ^7 + ( 8 0 ) * 0.85^0 * ( 1- 0.85 ) ^8
= 0.10521
d.
P( X = 2 ) = ( 8 2 ) * ( 0.85^2) * ( 1 - 0.85 )^6
= 0.00023

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