Mathematics: 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

Less than 2 = z 2Less than or equal to 8 = z ≤ 8 z ≤ 8z 2...

A continuous random variable X has a normal distribution

rachelkim999

21.01.2022

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

Step-by-step explanation:

We are given the following in the question:

Mean = 169

The variable X is normally distributed.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

P(x 180) = 0.17

$P( X 180) = P( z \displaystyle\frac{180 - 169}{\sigma})=0.17$

$= 1 -P( z \leq \displaystyle\frac{11}{\sigma})=0.17$

$=P( z \leq \displaystyle\frac{11}{\sigma})=0.83$

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

$\displaystyle\frac{11}{\sigma} = 0.954\\\\\sigma = 11.53$

We have to evaluate

P(x < 158)

$P( x < 158) = P( z < \displaystyle\frac{158- 169}{11.53}) = P(z < -0.954)$

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.

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