IXL - Find probabilities using the normal distribution I (Algebra 2 practice)

is a normally distributed random variable with mean 70 and standard deviation 8.

What is the probability that

is between 46 and 94?

Use the 0.68-0.95-0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

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Learn with an example

question

is a normally distributed random variable with mean 41 and standard deviation 2.

What is the probability that

is between 37 and 45?

Use the 0.68-0.95-0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

key idea

A random variable with a normal distribution is called normally

distributed.

For a normally distributed random variable

with mean

𝜇

and standard deviation

𝜎

(

𝜇

–

𝜎

𝜇

𝜎

)

≈

0.68

(

𝜇

–

𝜎

𝜇

𝜎

)

≈

0.95

(

𝜇

–

𝜎

𝜇

𝜎

)

≈

0.997

In other words:

The probability that
X
is within one standard deviation of the mean is
P
≈
0.68
.
The probability that
X
is within two standard deviations of the mean is
P
≈
0.95
.
The probability that
X
is within three standard deviations of the mean is
P
≈
0.997
.

solution

The probability that

is between 37 and 45 can be written as

(

)

Write 37 and 45 as the mean,

𝜇

plus or minus a multiple

of the standard deviation,

𝜎

𝜇

𝜎

–

So,

𝜇

–

𝜎

By a similar calculation,

𝜇

𝜎

This means

(

)

(

𝜇

–

𝜎

𝜇

𝜎

)

Since

is normally distributed:

(

𝜇

–

𝜎

𝜇

𝜎

)

≈

0.95

So,

(

)

(

𝜇

–

𝜎

𝜇

𝜎

)

≈

0.95

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