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Sampling Distribution of the Mean

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The Sampling Distribution of the Sample Mean

Let’s start at a familiar point: the sample mean, which we know as “x-bar”

Any random sample we might select from a population will, in general, have a different sample mean, right?

In other words, the sample mean varies from sample to sample...

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The Sample Mean is a Random Variable!

Imagine selecting many different samples from one population.

Sample 1

Sample 2Sample 3

Sample jSample k

•Unless there’s a miraculous coincidence,

we would expect each sample to have a

(slightly) different mean

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The Sample Mean is a Random Variable

So, x-bar varies from sample to sample.

Since the samples are random, x-bar varies due to a random process.

That makes x-bar a random variable!

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The Sample Mean is a Random Variable

So, to sum up so far: We can think of a descriptive statistic, like the

sample mean, as a random variable, since its value depends on the luck of the draw in random sampling.

Even though one sample has just one sample mean, this one mean is one of the many possible results of random sampling.

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So what is a “Sampling Distribution?”

The distribution of an “Ordinary Random Variable” has

• Shape• Center• Spread

The distribution of a descriptive statistic (like x-bar) has

• Shape• Center• Spread

Any random variable has a probability distribution or (if it’s continuous) a probability density function

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So what is a “Sampling Distribution?”

A sampling distribution is nothing more or less than a special case of a probability

distribution.

More specifically, if the random variable happens to be a sample statistic, we call its

distribution a Sampling Distribution

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Other Terminology

In a Sampling Distribution,

The shape is called the Sampling DistributionSampling Distribution (it’s often normal)

The center is still called the MeanMean

The spread is called the Standard ErrorStandard Error

The special terms are to keepyou from getting confused!?!

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Sampling Distribution of the Sample Mean: SHAPE

As we’ll soon see, if samples are large enough, the sampling distribution of the mean is approximately normal.

Like any normal distribution, it is defined by its mean and standard deviation, or standard errorstandard error

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Sampling Distribution of Two Dice

• A sampling distribution is created by looking at

• all samples of size n=2 (i.e. two dice) and their means…

• While there are 36 possible samples of size 2, there are only 11 values for x-bar, and some (e.g. x-bar = 3.5) occur more frequently than others (e.g. x-bar = 1).

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• The sampling distribution of x-bar is shown below:

Sampling Distribution of Two Dice…

P( ) 6/36

5/36

4/36

3/36

2/36

1/36

P(

)

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Compare…

• Compare the distribution of X…

• …with the sampling distribution of x-bar.

• As well, note that:

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Sampling Distribution of the Sample Mean: CENTER

The mean of the sampling distribution of the sample mean is simply the population mean,

.μ

In other words, the mean of the “average” sample should be the same

as the population mean, μ

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Sampling Distribution of the Sample Mean: SPREAD

The dispersion of the sampling distribution (in most cases) is given by this formula:

We call this the STANDARD ERROR OF

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Central Limit Theorem…

The sampling distribution of the mean of a random sample drawn from any population is approximately normal for a sufficiently large sample size.

The larger the sample size, the more closely the sampling distribution of x-bar will resemble a normal distribution.

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Central Limit Theorem…

If the population is normal, then x-bar is normally distributed for all values of n (n = the number of samples that x-bar is the sample of their means).

If the population is non-normal, then x-bar is approximately normal only for larger values of n.

In many practical situations, a sample size of 30 may be sufficiently large to allow us to use the normal distribution as an approximation for the sampling distribution of x-bar.

Central Limit TheoremCentral Limit Theorem

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Example 1

The foreman of a bottling plant has observed that the amount of soda in each “32-ounce” bottle is actually a normally distributed random variable, with a mean of 32.2 ounces and a standard deviation of 0.3 ounce.

If a customer buys one bottle, what is the probability that the bottle will contain more than 32 ounces?

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Example 1

We want to find P(X > 32), where x-bar is normally distributed and = 32.2 and = 0.3μ σ

“there is about a 75% chance that a single bottle of soda contains more than 32oz.”

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Example 2

The foreman of a bottling plant has observed that the amount of soda in each “32-ounce” bottle is actually a normally distributed random variable, with a mean of 32.2 ounces and a standard deviation of 0.3 ounce.

If a customer buys a carton of four bottles, what is the probability that the mean amount of the four bottles will be greater than 32 ounces?

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We want to find P(x-bar > 32), where X is normally distributed

with = 32.2 and = 0.3μ σThings we know:

1) X is normally distributed, therefore so will x-bar.

2) = 32.2 oz.

3)

Example 2

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Example 2

If a customer buys a carton of four bottles, what is the probability that the mean amount of the four bottles will be greater than 32 ounces?

“There is about a 91% chance the mean of the four bottles will exceed 32oz.”

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Graphically Speaking…

what is the probability that one bottle will contain more than 32 ounces?

what is the probability that the mean of four bottles will exceed 32 oz?

mean=32.2

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Example 3

The dean of the School of Business claims that the average salary of the school’s graduates one year after graduation is $800 per week with a standard deviation of $100.

A second-year student would like to check whether the claim about the mean is correct. He does a survey of 25 people who graduated one year ago and determines their weekly salary. He discovers the sample mean to be $750.

To interpret his finding he needs to calculate the probability that a sample of 25 graduates would have a mean of $750 or less when the population mean is $800 and the standard deviation is $100.

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• We want to compute

• Although X is likely skewed it is likely that x-bar is normally distributed. The mean of x-bar is

• The standard deviation is

Example 3

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Example 3

The probability of observing a sample mean as low as $750 when the population mean is $800 is extremely small. Because the event is quite unlikely, we would conclude that the dean’s claim is not justified.

Example 4Example 4

Cola bottles are filled so that Cola bottles are filled so that contents X have a normal distribution contents X have a normal distribution with mean = 298 ml and standard with mean = 298 ml and standard deviation sigma = 3 ml.deviation sigma = 3 ml.

What proportion of bottles have less What proportion of bottles have less than 295 ml?than 295 ml?

Ans: P(X < 295) = P(Z < –1) = 0.1586Ans: P(X < 295) = P(Z < –1) = 0.1586

Example 4Example 4What is the probability of the average of What is the probability of the average of a six pack of bottles being less than 295 a six pack of bottles being less than 295 ml?ml?Ans: P(x-bar < 295) = Ans: P(x-bar < 295) =

Example 4Example 4

P(X < 295) = .15, but P(x-bar < 295) = 0.007P(X < 295) = .15, but P(x-bar < 295) = 0.007

As the sample size increases, the variation in the As the sample size increases, the variation in the distribution decreases so that a value like 295 ml is very distribution decreases so that a value like 295 ml is very difficult and rare to occur in an average of a six pack or difficult and rare to occur in an average of a six pack or more of bottles, but could quite easily occur in a single more of bottles, but could quite easily occur in a single bottle.bottle.

Big point: averages have less variation than individual Big point: averages have less variation than individual observations.observations.

Assignment

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