Question

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 820 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?

*(Round the values of z to 2 decimal places. Round the intermediate values to 4 decimal places. Round your answer to 4 decimal places.)
**(Round the values of z to 2 decimal places. Round the intermediate values to 4 decimal places. Round your answer to 5 decimal places.)

Answer 1

a)

Let X is a random variable shows the number of consumers claim that how a company handles a crisis when at fault is an influence in their buying decisions. Here X has binomial distribution with following parameter

n = 1100, p = 0.73

Since np = 803 and n(1-p) = 297 both are greater than 5 so we can use normal approximation here.

Using normal approximation, X has approximately normal distribution with mean and SD as follows:

The z-score for X = 820 +0.5 = 820.5 is

The probability that more than 820 consumers claim that how a company handles a crisis when at fault is an influence in their buying decisions is

P(X > 820) = P(X > 820.5) = 1- P(z <= 1.19) = 1 - 0.883 = 0.1170

b)

Let X is a random variable shows the number of consumers claim that how a company handles a crisis when at fault is an influence in their buying decisions. Here X has binomial distribution with following parameter

n = 1100, p = 0.96

Since np = 1056 and n(1-p) = 44 both are greater than 5 so we can use normal approximation here.

Using normal approximation, X has approximately normal distribution with mean and SD as follows:

The z-score for X = 1030 -0.5 = 1029.5 is

The probability that fewer than 1,030 consumers claim that quality of product is an influence in their buying decisions is

P(X < 1030) = P(X < 1029.5) = P(z < -4.08) = 0.00002

Excel function used "=NORMSDIST(-4.08)"

c)

Here we have

n=1100, p = 0.85

The sampling distribution of sample proportion will be approximately normal with mean

and standard deviation

The z-score for is

The z-score for is

The probability that between 82% and 83% of consumers claim that how a company handles complaints is an influence in their buying decisions is

Answer: 0.0295