Question

A company's sales force makes 400 sales calls, with + 0.25 probability that a sale will be made on a call. What is the probability that greater than 55 (exclusive) but less than 75 (exclusive) sales will be made? Enter your answer as a decimal value, rounded to 4 decimal places.

Answer 1

0.0016 probability that greater than 55 (exclusive) but less than 75 (exclusive) sales will be made.

Explanation:

We use the normal approximation to the binomial distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Normal probability distribution

Problems of normally distributed distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that
`\mu = E(X)`,
`\sigma = √(V(X))`.

A company's sales force makes 400 sales calls, with 0.25 probability that a sale will be made on a call.

This means that
`n = 400, p = 0.25`

Mean and standard deviation:

`\mu = E(X) = np = 400*0.25 = 100`

`\sigma = √(V(X)) = √(np(1-p)) = √(400*0.25*0.75) = √(75)`

What is the probability that greater than 55 (exclusive) but less than 75 (exclusive) sales will be made?

Using continuity correction, this is
`P(55+0.5 \leq X \leq 75-0.5) = P(55.5 \leq X \leq 74.5)`, which is the p-value of Z when X = 74.5 subtracted by the p-value of Z when X = 55.5.

X = 74.5

`Z = (X - \mu)/(\sigma)`

`Z = (74.5 - 100)/(√(25))`

`Z = -2.94`

`Z = -2.94` has a p-value of 0.0016

X = 55.5

`Z = (X - \mu)/(\sigma)`

`Z = (55.5 - 100)/(√(25))`

`Z = -5.14`

`Z = -5.14` has a p-value of 0

0.0016 - 0 = 0.0016

0.0016 probability that greater than 55 (exclusive) but less than 75 (exclusive) sales will be made.