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

is a normally distributed random variable with mean 43 and standard deviation 19.

What is the probability that

is between 24 and 62?

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

ref_doc_title.

Learn with an example

question

is a normally distributed random variable with mean 47 and standard deviation 15.

What is the probability that

is between 2 and 92?

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 2 and 92 can be written as

(

)

Write 2 and 92 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

)

≈

0.997

So,

(

)

(

𝜇

–

𝜎

𝜇

𝜎

)

≈

0.997

Looking for more? Try this:

$Confetti$

Excellent!

Now entering the Challenge Zone—are you ready?

Questions
answeredQuestions

Time
elapsedTime

hr	min	sec
00	00	04

SmartScore
out of 100
IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)! Learn more

Teacher tools

Work it out

Not feeling ready yet? This can help:

DD.7Standard deviation of random variables

Excellent! You got that right!

Jumping to level 1 of 1

Excellent!