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

Answer:

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

Step-by-step 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:

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

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 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

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

$Z = \frac{118.8 - 100}{10}$

$Z = 1.88$

$Z = 1.88$ has a pvalue of 0.97

X = 100

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

$Z = \frac{100 - 100}{10}$

$Z = 0$

$Z = 0$ 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