Question

1. Suppose you have a variable X~N(8, 1.5). What is the probability that you have values between (6.5, 9.5)

Answer 1

0.6826 = 68.26% probability that you have values in this interval.

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.

X~N(8, 1.5)

This means that
`\mu = 8, \sigma = 1.5`

What is the probability that you have values between (6.5, 9.5)?

This is the p-value of Z when X = 9.5 subtracted by the p-value of Z when X = 6.5. So

X = 9.5

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

`Z = (9.5 - 8)/(1.5)`

`Z = 1`

`Z = 1` has a p-value of 0.8413.

X = 6.5

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

`Z = (6.5 - 8)/(1.5)`

`Z = -1`

`Z = -1` has a p-value of 0.1587

0.8413 - 0.1587 = 0.6826

0.6826 = 68.26% probability that you have values in this interval.