Question

Given a normal distribution with mean 100 and standard deviation​ 10, find the area under the normal curve from the mean to 118.8.

Answer 1

The area under the normal curve from the mean to 118.8. is 0.47

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.

In this question, we have that:

`\mu = 100, \sigma = 10`

Find the area under the normal curve from the mean to 118.8.

This is the pvalue of Z when X = 118.8 subtracted by the pvalue of Z when X = 100.

X = 118.8

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

`Z = (118.8 - 100)/(10)`

`Z = 1.88`

`Z = 1.88` has a pvalue of 0.97

X = 100

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

`Z = (100 - 100)/(10)`

`Z = 0`

`Z = 0` has a pvalue of 0.5

0.97 - 0.5 = 0.47

The area under the normal curve from the mean to 118.8. is 0.47