Question

Consider a normal distribution curve where the middle 85 % of the area under the curve lies above the interval ( 8 , 14 ). Use this information to find the mean, \mu , and the standard deviation, \sigma , of the distribution.

Answer 1

`\mu = 11`

`\sigma = 2.08`

Explanation:

Problems of normally distributed samples are 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.

Middle 85%.

Values of X when Z has a pvalue of 0.5 - 0.85/2 = 0.075 to 0.5 + 0.85/2 = 0.925

Above the interval (8,14)

This means that when Z has a pvalue of 0.075, X = 8. So when
`Z = -1.44, X = 8`. So

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

`-1.44 = (8 - \mu)/(\sigma)`

`8 - \mu = -1.44\sigma`

`\mu = 8 + 1.44\sigma`

Also, when X = 14, Z has a pvalue of 0.925, so when
`X = 8, Z = 1.44`

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

`1.44 = (14 - \mu)/(\sigma)`

`14 - \mu = 1.44\sigma`

`1.44\sigma = 14 - \mu`

Replacing in the first equation

`\mu = 8 + 1.44\sigma`

`\mu = 8 + 14 - \mu`

`2\mu = 22`

`\mu = (22)/(2)`

`\mu = 11`

Standard deviation:

`1.44\sigma = 14 - \mu`

`1.44\sigma = 14 - 11`

`\sigma = (3)/(1.44)`

`\sigma = 2.08`