Question

Suppose x is a normally distributed random variable with a mean of 22 and a standard deviation of 5. The probability that x is less than 9.7 is _____.

Answer 1

Answer:

The probability that x is less than 9.7 is 0.0069 = 0.69%

Step-by-step explanation:

Problems of normally distributed samples are 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:

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

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 problem, we have that:

$\mu = 22, \sigma = 5$

The probability that x is less than 9.7 is _____.

This is the pvalue of Z when X = 9.7. So

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

$Z = \frac{9.7 - 22}{5}$

$Z = -2.46$

$Z = -2.46$ has a pvalue of 0.0069

The probability that x is less than 9.7 is 0.0069 = 0.69%