IXL - Find probabilities using the normal distribution II (Precalculus practice)

is a normally distributed random variable with mean 85 and standard deviation 3.

What is the probability that

is between 77 and 93?

Write your answer as a decimal rounded to the nearest thousandth.

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Learn with an example

question

is a normally distributed random variable with mean 6 and standard deviation 17.

What is the probability that

is between 1 and 11?

Write your answer as a decimal rounded to the nearest thousandth.

key idea

A random variable with a normal distribution is called normally

distributed.

The standard normal distribution is the normal distribution with mean 0 and standard deviation 1. A random variable with the standard normal distribution is called a standard normal

variable.

is a normally distributed random variable with mean

𝜇

and standard deviation

𝜎

then the distribution of

–

𝜇

𝜎

is the standard normal distribution.

For a specific value

–

𝜇

𝜎

is called the

-value.

-value

is a specific value of

solution

Use the random variable

–

𝜇

𝜎

Since

is normally distributed,

has the standard normal distribution.

To find the probability that

is between 1 and 11, first find the

-values

that correspond to the

values of 1 and 11. Call them

and

Then, find the probability that

is between

and

Find the

-values,

and

Use the equation

–

𝜇

𝜎

Plug each value of

into the equation along with

𝜇

and

𝜎

–

𝜇

𝜎

–

0.294117…

–

𝜇

𝜎

–

0.294117…

The two

-values

are

–

0.294117…

and

0.294117…

Find the probability that

is between

and

Since

is a standard normal variable, use a calculator to find the probability that it is between

–

0.294117…

and 0.294117….

(

–

0.294117…

)

0.231331…

≈

Round to the nearest thousandth

0.231

Round to the nearest thousandth

So,

(

)

(

–

0.294117…

)

≈

0.231

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DD.7Standard deviation of random variables

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