Question

The speed with which utility companies can resolve problems is very important. GTC, the Georgetown Telephone Company reports it can resolve customer problems the same day they are reported in 70% of the cases. Suppose the 12 cases reported today are representative of all complaints

a-1. How many of the problems would you expect to be resolved today? (Round your answer to 2 decimal places.)
a-2. What is the standard deviation? (Round your answer to 4 decimal places.)
b. What is the probability 9 of the problems can be resolved today? Round your answer to 4 decimal places.)
c. What is the probability 9 or 10 of the problems can be resolved today? (Round your answer to 4 decimal places.) Probability decimal places)
d. What is the probability more than 7 of the problems can be resolved today?

Answer 1

8.40 ; 1.5875; 0.2397 ; 0.4075 ; 0.7237

Explanation:

Given that:

p = 70% = 0.7

Sample size, n = 12

A.) number of problems expected to be resolved:

E(x) = np = 12 * 0.7 = 8.40 (2 decimal place)

Standard deviation :

Sqrt(n*p*q)

q = 1 - p ; q = 1 - 0.7 = 0.3

Standard deviation = sqrt(12 * 0.7 * 0.3)

Standard deviation = sqrt(2.52)

Standard deviation = 1.58745

Standard deviation = 1.5875 (4 decimal places)

B.) 9 of the problems will be resolved :

P(x = 9)

Using binomial distribution formula :

P(x =x) = nCx * p^x * (1 - p)^(n - x)

P(x = 9) = 12C9 * 0.7^9 * 0.3^3

P(x = 9) = 220 * 0.040353607 * 0.027

P(x = 9) = 0.23970042558

P(x = 9) = 0.2397

C.) p(9 or 10) = p(x =9) + p(x = 10)

P(x = 9) = 0.2397

P(x = 10) = 12C10 * 0.7^10 * 0.3^2

P(x = 10) = 66 * 0.7^10 * 0.3^2

p(x = 9 or x = 10) = 0.2397 + 0.1678

p(x = 9 or x = 10) = 0.4075

D) probability that more than 7 will be resolved :

P(x > 7) = p(x =8) + p(x =9) + p(x = 10) + p(x = 11) + p(x = 12)

We could make use of a binomial probability calculator in other to save time :

P(x > 7) = 0.7237