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

Answer:

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

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{9.5 - 8}{1.5}$

$Z = 1$

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

X = 6.5

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{6.5 - 8}{1.5}$

$Z = -1$

$Z = -1$ 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.