IXL - The Central Limit Theorem (Precalculus practice)

Samples are drawn from a population with mean 152 and standard deviation 42. Each sample has 345 randomly and independently chosen elements.

Use the Central Limit Theorem to estimate the probability that a sample mean is between 150 and 154.

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

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Learn with an example

question

Samples are drawn from a population with mean 138 and standard deviation 29. Each sample has 202 randomly and independently chosen elements.

Use the Central Limit Theorem to estimate the probability that a sample mean is between 135 and 141.

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

key idea

Distribution of sample means

The mean of a sample from a population is called the sample

mean.

Think about samples of

randomly and independently chosen elements from the same population. A sample mean can be computed for each sample. The computed sample means are values of a random variable, usually called

The probability distribution of

depends on the sample size,

and the distribution of the population. If the population has mean

𝜇

and standard deviation

𝜎

the following are the mean and standard deviation for the distribution of sample means:

𝜇

𝜎

Central Limit Theorem

As the sample size increases, the distribution of sample means approaches a normal distribution, even if the population is not normally distributed.

So, if the sample size

is large, the distribution of

can be approximated by a normal distribution with mean

𝜇

and standard deviation

𝜎

Normally distributed random variables

is a normally distributed random variable with mean

𝜇

and standard deviation

𝜎

then the distribution of

–

𝜇

𝜎

is the standard normal distribution with mean 0 and standard deviation 1.

For a specific value

of the random variable

–

𝜇

𝜎

is called the

-value.

-value

is a specific value of the random variable

solution

The probability that a sample mean is between 135 and 141 is the probability that the random variable

is between 135 and 141. Approximate the distribution of

then calculate the probability.

start of large bold text, Use the Central Limit Theorem to approximate the distribution of x., end of large bold text

Since

202

is large, you can use the Central Limit Theorem to treat

as a normally distributed random variable. To completely determine its distribution, find its mean and standard deviation.

Find

𝜇

The distribution of sample means has the same mean as the population the samples are taken from. So,

𝜇

138

Find

𝜎

Each sample has 202 elements, so

202

The population has a standard deviation of 29, so

𝜎

Use these values in the formula for the standard deviation of the distribution of sample means.

𝜎

202

2.040432…

start of large bold text, Calculate the probability that x is between 135 and 141., end of large bold text

Use the random variable

–

𝜇

𝜎

Since you are treating

as a normally distributed random variable,

has the standard normal distribution.

To find the probability that

is between 135 and 141, first find the

-values

that correspond to the

-values

of 135 and 141. 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

138

and

𝜎

2.040432…

135

–

𝜇

𝜎

135

–

138

2.040432…

–

2.040432…

–

1.470276…

141

–

𝜇

𝜎

141

–

138

2.040432…

1.470276…

The

-values

are

–

1.470276…

and

1.470276…

Find the probability that

is between

and

Since

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

–

1.470276…

and 1.470276….

(

–

1.470276…

)

0.858513…

≈

Round to the nearest thousandth

0.859

Round to the nearest thousandth

Using the Central Limit Theorem, the probability that a sample mean is between 135 and 141 is approximately 0.859.

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Work it out

Not feeling ready yet? These can help:

EE.6Find values of normal variables

EE.7Distributions of sample means

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