Question

5. X is a normally distributed random variable with a mean of 8 and a standard deviation of 4. This can be written as N(8,4). The probability that X is between 1.48 and 15.56 is _____________. Write out the probability notation for this question.

Answer 1

The probability that X is between 1.48 and 15.56 is
`P(1.48 \leq X \leq 15.56) = 0.919`

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 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.

X is a normally distributed random variable with a mean of 8 and a standard deviation of 4.

This means that
`\mu = 8, \sigma = 4`

The probability that X is between 1.48 and 15.56

This is the pvalue of Z when X = 15.56 subtracted by the pvalue of Z when X = 1.48. So

X = 15.56

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

`Z = (15.56 - 8)/(4)`

`Z = 1.89`

`Z = 1.89` has a pvalue of 0.9706

X = 1.48

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

`Z = (1.48 - 8)/(4)`

`Z = -1.63`

`Z = -1.63` has a pvalue of 0.0516

0.9706 - 0.0516 = 0.919

Write out the probability notation for this question.

`P(1.48 \leq X \leq 15.56) = 0.919`

The probability that X is between 1.48 and 15.56 is
`P(1.48 \leq X \leq 15.56) = 0.919`