sampling distributions. example take random sample of students. ask “how many courses did you...

Sampling distributions

Example

• Take random sample of students.

• Ask “how many courses did you study for this past weekend?”

• Calculate a statistic, say, the sample mean.

Sample 1: 1 0 2 Mean = 1.0

Sample 2: 1 1 4 Mean = 2.0

Situation

• Different samples produce different results.

• Value of a statistic, like mean or proportion, depends on the particular sample obtained.

• But some values may be more likely than others.

• A “sampling distribution” is a probability distribution of a statistic. It indicates the likelihood of getting certain values.

Sampling distribution of meanIF:

• data are normally distributed with mean and standard deviation , and

• random samples of size n are taken, THEN:

The sampling distribution of the sample means is also normally distributed.

The mean of all of the sample means is .

The standard deviation of the sample means (“standard error of the mean”) is /sqrt(n).

Example

• Adult nose length is normally distributed with mean 45 mm and standard deviation 6 mm.

• Take random samples of n = 4 adults.

• Then, sample means are normally distributed with mean 45 mm and standard error 3 mm [from 6/sqrt(4) = 6/2].

Using empirical rule...

• 68% of samples of n=4 adults will have an average nose length between 42 and 48 mm.

• 95% of samples of n=4 adults will have an average nose length between 39 and 51 mm.

• 99% of samples of n=4 adults will have an average nose length between 36 and 54 mm.

What happens if we take larger samples?

• Adult nose length is normally distributed with mean 45 mm and standard deviation 6 mm.

• Take random samples of n = 36 adults.

• Then, sample means are normally distributed with mean 45 mm and standard error 1 mm [from 6/sqrt(36) = 6/6].

Again, using empirical rule...

• 68% of samples of n=36 adults will have an average nose length between 44 and 46 mm.

• 95% of samples of n=36 adults will have an average nose length between 43 and 47 mm.

• 99% of samples of n=36 adults will have an average nose length between 42 and 48 mm.

• So … the larger the sample, the less the sample averages vary.

What happens if data are not normally distributed?

The Central Limit Theorem tells us ...

Central Limit Theorem• Even if data are not normally distributed,

as long as you take “large enough” samples, the sample averages will at least be approximately normally distributed.

• Mean of sample averages is still .• Standard error of sample averages is still

/sqrt(n).• In general, “large enough” means more than

30 measurements.

Big Deal?

• Knowing the distribution of sample means allows us to make decisions about the value of a population mean.

• Let’s look at an application …