SAMPLING DISTRIBUTIONS

2) How else can we estimate the population mean value, if we can not take very large samples for our study?

e.g. What can we say about the estimate of mean

from say 10 subjects as an estimate of μ?

Here are the odor thresholds (micrograms of DMS per

liter of wine) for 10 randomly chosen subjects:

28 40 28 33 20 31 29 27 17 21

• How well would our mean value from this sample estimate the true parameter value?

SAMPLING DISTRIBUTIONS

What would happen if we took many samples of

10 subjects from the population?

• Steps:

1. Take a large number of samples of size 10 from the population

2. Calculate the sample mean for each sample

3. Make a histogram of the mean values4. Examine the distribution displayed in

the histogram for shape, center, and spread, as well as outliers and other deviations

Properties of sampling distributions

Mean and standard deviation of sampling distribution of a mean value

• sampling distribution of will be centered at

• is an unbiased estimator of

x

x

Mean and standard deviation of sampling distribution of a mean value

• is an unbiased estimator of

– In repeated sampling, sometimes overestimates, sometimes underestimates

– No systematic tendency of either outcome exists

– Lack of bias or favoritism

– Correct on the average in many samples

x

xx

Mean and standard deviation of sampling distribution of a mean value

• How can we maximize our chances of having the estimator fall to the parameter in as many samples as possible?

Mean and standard deviation of sampling distribution of a mean value

• How can we maximize our chances of having the estimator fall to the parameter in as many samples as possible?

– Determined by spread of sampling distribution

– If individual observations has standard deviation then sample means x from samples of size n have s =

– Averages are less variable than individuals

– values of s get smaller as we take larger samples (n is larger)

• Results of large samples are less variable than results of small samples

• Sample means from large samples are trusted estimators of

n

Mean and standard deviation of sampling distribution of a mean value

Problem:

• Knowing that s = , how much do we need to

increase the sample size in order to cut the standard

deviation in half?

n

Sampling distribution of a sample mean

• Shape of sampling distribution of depends on the population distribution.

– If population distribution is normal, then so is the distribution of sample mean.

• If a population has the N(,) distribution, then the sample mean x of n independent observations has the N(, ) distribution.

n

x

Central Limit Theorem

• What happens when the population distribution is not normal?

• As we increase the number of observations that we use to draw our sample, the sampling distribution of x changes its shape

N = 1 N = 2

N = 10 N = 25

Central Limit Theorem

• Draw a simple random sample of size n from any population with mean and standard deviation .

• When n is large, the sampling distribution of the sample mean x is approximately normal N(, )

• Applies to sum or averages

n