Question

An investment service is currently recommending the purchase of shares of Dollar Department Store selling at $18 per share. The price is approximately normally distributed with a mean of 20 and a standard deviation of 2. What is the probability that in a year the shares will be selling: Between $21 and $24

Answer 1

0.2857 = 28.57% probability that in a year the shares will be selling between $21 and $24

Explanation:

When the distribution is normal, we use 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.

The price is approximately normally distributed with a mean of 20 and a standard deviation of 2.

This means that
`\mu = 20, \sigma = 2`

What is the probability that in a year the shares will be selling between $21 and $24

This is the pvalue of Z when X = 24 subtracted by the pvalue of Z when X = 21. So

X = 24:

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

`Z = (24 - 20)/(2)`

`Z = 2`

`Z = 2` has a pvalue of 0.9772

X = 21:

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

`Z = (21 - 20)/(2)`

`Z = 0.5`

`Z = 0.5` has a pvalue of 0.6915

0.9772 - 0.6915 = 0.2857

0.2857 = 28.57% probability that in a year the shares will be selling between $21 and $24