Question

A continuous random variable X has a normal distribution with mean 169. The probability that X takes a value greater than 180 is 0.17. Use this information and the symmetry of the density function to find the probability that X takes a value less than 158

Answer 1

0.17 is the probability that X takes a value less than 158.

Explanation:

We are given the following in the question:

Mean = 169

The variable X is normally distributed.

Formula:

`z_(score) = \displaystyle(x-\mu)/(\sigma)`

`P(x > 180) = 0.17`

`P( X > 180) = P( z > \displaystyle(180 - 169)/(\sigma))=0.17`

`= 1 -P( z \leq \displaystyle(11)/(\sigma))=0.17`

`=P( z \leq \displaystyle(11)/(\sigma))=0.83`

Calculation the value from standard normal z table, we have,

`\displaystyle(11)/(\sigma) = 0.954\\\\\sigma = 11.53`

We have to evaluate

P(x < 158)

`P( x < 158) = P( z < \displaystyle(158- 169)/(11.53)) = P(z < -0.954)`

Calculation the value from standard normal z table, we have,

`P(x < 158) = 0.170`

0.17 is the probability that X takes a value less than 158.