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.

[tex]\text{a. What is the probability that more than 810 consumers claim that how a company } \\ \\ \text { handles a crisis when at fault is an influence in their buying decisions?}[/tex][tex]\text{b. What is the probability that fewer than 1,030 consumers claim that quality} \\ \\ \text{ of product is an influence in their buying decisions?}[/tex]

From the given information:

the sample size = 1100

P = 73% = 0.73

Sample proportion [tex]\hat p = \dfrac{810}{1100} = 0.7364[/tex]

The Z test statistics can be computed as:

[tex]Z = \dfrac{\hat p - p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]

[tex]Z = \dfrac{0.7364 - 0.73}{\sqrt{\dfrac{0.73(1-0.73)}{1100}}}[/tex]

[tex]Z = \dfrac{0.0064}{\sqrt{\dfrac{0.1971}{1100}}}[/tex]

[tex]Z =0.478[/tex]

a) Now the required probability of more than 810 claims is:

[tex]P(X> 810) = 1 - P(X<810) \\ \\ \implies 1 - ( X< 0.478) \\ \\ \implies 1 - 0.68367 \\ \\ = \mathbf{0.3163}[/tex]

b) The probability of less than 1030 consumers is:

Probability of quality product influencers = 96% = 0.96

Sample proportion [tex]\hat p = \dfrac{1030}{1100}[/tex] = 0.9364

The Z test Statistics is:

[tex]Z = \dfrac{\hat p - p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]

[tex]Z = \dfrac{0.9364 - 0.96}{\sqrt{\dfrac{0.96(1-0.96)}{1100}}}[/tex]

[tex]Z = -3.994[/tex]

[tex]P(X < 1030) = 1 - P(3.99) \\ \\ \implies 1 - 0.99999669 \\ \\ \implies \mathbf{0.00003 }[/tex]