In: Statistics and Probability
A survey of 2645 consumers by DDB Needham Worldwide of Chicago for public relations agency Porter/Novelli showed that how a company handles a crisis when at fault is one of the top influences in consumer buying decisions,with 73% claiming it is an influence. Quality of product was the number one influence, with 96% of consumers stating that quality influences their buying decisions. How a company handles complaints was number two, with 85% of consumers reporting it as an influence in their buying decisions. Suppose a random sample of 1,100 consumers is taken and each is asked which of these three factors influence their buying decisions.
a. What is the probability that more than 830 consumers claim that how a company handles a crisis when at fault is an influence in their buying decisions?
b. What is the probability that fewer than 1,030 consumers claim that quality of product is an influence in their buying decisions?
c. What is the probability that between 82% and 83% of consumers claim that how a company handles complaints is an influence in their buying decisions?
Solution
Back-up Theory
If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of
trials and p = probability of one success, then, probability mass function (pmf) of X is given by
p(x) = P(X = x) = (nCx)(px)(1 - p)n – x, …………..........................................................................................………..(1)
[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST].(1a)
X ~ B(n, p) np ≥ 5 and np(1 - p) ≥ 5, (X – np)/√{np(1 - p)} ~ N(0, 1) [approximately].............................................. (2)
Probability values for the Standard Normal Variable, Z, can be directly read off from Standard Normal Tables … (2a)
or can be found using Excel Function: Statistical, NORMSDIST(z) which gives P(Z ≤ z) …..................................(2b)
Now to work out the solution,
Let
X = number of consumers out of a sample of 1100 consumers who claim that how a
company handles a crisis when at fault is an influence in their buying decisions ......................................... (3a)
Y = number of consumers out of a sample of 1100 consumers who claim that quality
of product is an influence in their buying decisions ........................................................................................ (3b)
U = number of consumers out of a sample of 1100 consumers who claim that how a
company handles complaints is an influence in their buying decisions ........................................................ (3c)
Then,
X ~ B(1100, 0.73) [0.73 = 73%] ........................................................................................................................... (4a)
Y ~ B(1100, 0.96) [0.96 = 96%] .......................................................................................................................... (4b)
X ~ B(1100, 0.85) [0.85 = 85%] .......................................................................................................................... (4c)
Part (a)
Probability that more than 830 consumers claim that how a company handles a crisis when at fault is an influence in their buying decisions
= P(X > 830)
= P[Z > (830 – 803)/√{216.81)}] [vide (3a), (4a) and (2)]
= P(Z > 1.8336)
= 0.0334 [vide (2b)] Answer 1
Part (b)
Probability that fewer than 1,030 consumers claim that quality of product is an influence in their buying decisions
= P(Y < 1030)
= P[Z < (1030 – 1056)/√{42.24)}] [vide (3b), (4b) and (2)]
= P(Z < - 5.2737)
= 0 [vide (2b)] Answer 2
Part (c)
Probability that between 82% and 83% of consumers claim that how a company handles complaints is an influence in their buying decisions
= P(902 < U < 913)
= P[(902 – 935)/√{140.25)} < Z < (913 – 935)/√{140.25)}] [vide (3c), (4c) and (2)]
= P(- 2.5176 < Z < - 1.8576)
= P(Z < - 1.8576) - P(Z < - 2.5176)
= 0.0316 – 0.0059 [vide (2b)]
= 0.0257 Answer 3
DONE