Question

A company services home air conditioners. It is known that times for service calls follow a normal distribution with a mean of 75 minutes and a standard deviation of 15 minutes. A random sample of twelve service calls is taken. What is the probability that exactly eight of them take more than 93.6 minutes

Answer 1

The probability that exactly eight of them take more than 93.6 minutes is 5.6015
`* 10^(-6)` .

Explanation:

We are given that it is known that times for service calls follow a normal distribution with a mean of 75 minutes and a standard deviation of 15 minutes.

A random sample of twelve service calls is taken.

So, firstly we will find the probability that service calls take more than 93.6 minutes.

Let X = times for service calls.

So, X ~ Normal(
`\mu=75,\sigma^(2) =15^(2)`)

The z-score probability distribution for the normal distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = mean time = 75 minutes

`\sigma` = standard deviation = 15 minutes

Now, the probability that service calls take more than 93.6 minutes is given by = P(X > 93.6 minutes)

P(X > 93.6 min) = P(
`(X-\mu)/(\sigma)` >
`(93.6-75)/(15)` ) = P(Z > 1.24) = 1 - P(Z
`\leq` 1.24)

= 1 - 0.8925 = 0.1075

The above probability is calculated by looking at the value of x = 1.24 in the z table which has an area of 0.8925.

Now, we will use the binomial distribution to find the probability that exactly eight of them take more than 93.6 minutes, that is;

`P(Y = y) = \binom{n}{r}* p^(r) * (1-p)^(n-r) ; y = 0,1,2,3,.........`

where, n = number of trials (samples) taken = 12 service calls

r = number of success = exactly 8

p = probability of success which in our question is probability that

it takes more than 93.6 minutes, i.e. p = 0.1075.

Let Y = Number of service calls which takes more than 93.6 minutes

So, Y ~ Binom(n = 12, p = 0.1075)

Now, the probability that exactly eight of them take more than 93.6 minutes is given by = P(Y = 8)

P(Y = 8) =
`\binom{12}{8}* 0.1075^(8) * (1-0.1075)^(12-8)`

=
`495 * 0.1075^(8) * 0.8925^(4)`

= 5.6015
`* 10^(-6)` .