Question

The speed with which utility companies can resolve problems is very important. GTC, Georgetown Telephone Company, reports it can resolve customer problems the same day they are reported in 80% of the cases. Suppose the 15 cases reported today are representative of all complaints. What is the standard deviation? What is the probability 7 of the problems can be resolved today? What is the probability 7 or 8 of the problems can be resolved today? What is the probability more than 10 of the problems will be resolved today?

Answer 1

Part 1: 1.5492

Part 2: 0.0035

Part 3: 0.0173

Part 4: 0.8358

Explanation:

Number of cases, n = 15

Probability they can be resolved same day = 0.8 [p=0.8]

Part 1:

What is the Standard Deviation?

The formula is
`SD=√(np(1-p))`

Plugging in we get:

`SD=√(np(1-p)) \\SD=√((15)(0.8)(1-0.8))\\ SD=1.5492`

Standard Deviation is 1.5492

Part 2:

What is the probability 7 of the problems can be resolved today?

Since this is binomial probability, the formula is:

`P(x=r)=nCr*p^(r)*(1-p)^(n-r)`

Here, r = 7, so we plug in what we know and get out answer:

`P(x=r)=nCr*p^(r)*(1-p)^(n-r)\\P(x=7)=15C7*(0.8)^(7)*(1-0.8)^(15-7)\\P(x=7)=0.0035`

THe probability is 0.0035

Part 3:

"OR" in probability means "+", so we add

P(x=7) + P(x=8)

We already know P(x=7), we just find P(x=8) now.

Replacing r = 7 with r = 8 into the formula above, we get:

`P(x=8)=15C8*(0.8)^(8)*(1-0.8)^(15-8)\\P(x=8)=0.0138`

The probability is 0.0138

Hence P(x=7) + P(x=8) = 0.0035 + 0.0138 = 0.0173

Part 4:

More than 10 problems means P(x=11) + P(x=12) +....P(x=15)

we have to plug in 11,12,13,14,15 into r of the formula individually and find each and sum them.

The same process.

Basically it is going to be:

P(x>10) = P(x=11) + P(x=12) + P(x=13) + P(x=14) + P(x=15)

The answer after substituting and summing would be 0.8358 [using binomial calculator and table for ease]