Question

The Arc Electronic Company had an income of 59 million dollars last year. Suppose the mean income of firms in the same industry as Arc for a year is 45 million dollars with a standard deviation of 7 million dollars. If incomes for this industry are distributed normally, what is the probability that a randomly selected firm will earn more than Arc did last year

Answer 1

0.0228 = 2.28% probability that a randomly selected firm will earn more than Arc did last year

Explanation:

Normal Probability Distribution:

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

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Suppose the mean income of firms in the same industry as Arc for a year is 45 million dollars with a standard deviation of 7 million dollars

This means that
`\mu = 45, \sigma = 7`

What is the probability that a randomly selected firm will earn more than Arc did last year?

Arc earned 59 million, so this is 1 subtracted by the pvalue of Z when X = 59.

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

`Z = (59 - 45)/(7)`

`Z = 2`

`Z = 2` has a pvalue of 0.9772

1 - 0.9772 = 0.0228

0.0228 = 2.28% probability that a randomly selected firm will earn more than Arc did last year