• Chapter 7, Sample Distribution

–A sampling distribution is a distribution of all of the possible values of a statistic (say sample mean) for a given size sample selected from a population.

• Sample Distribution of the Mean is an Unbiased Estimate of the Population Mean

–If all possible samples of a certain size, n, are selected from a population, the mean of these sample means (the grand mean) would be equal to the population mean.

μμX

• Assume there is a population of size N=4

• Random variable, X, is age of individuals

• Values of X: 18, 20, 22, 24 (years)

• The population mean and standard deviation are:

214

24222018

N

Xμ i

2.236N

μ)(Xσ

2i

18 20 22 24 A B C D

Uniform Distribution

P(x)

Now consider all possible samples of size n=2 drawn from this population

1st 2nd Observation Obs 18 20 22 24

18 18,18 18,20 18,22 18,24

20 20,18 20,20 20,22 20,24

22 22,18 22,20 22,22 22,24

24 24,18 24,20 24,22 24,24

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

16 possible samples (sampling with replacement)

16 Sample Means

• Sampling Distribution of All Sample Means

18 19 20 21 22 23 24

P(X)

(no longer uniform)

2116

24211918

N

Xμ i

X

1.5816

21)-(2421)-(1921)-(18

N

)μX(σ

222

2

Xi

X

PopulationN = 4, 2.236σ 21μ

Sample Means Distributionn = 2, 1.58σ 21μ

XX

Notice:

• Why std. Dev. Of the means distribution is smaller than that of the population?

Reasons:

• Different samples of the same size from the same population will yield different sample means

• A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:

n

σσ

X

•Note that the standard error of the mean decreases and the distribution becomes less dispersed as the sample size increases (see page 235)

• If a population is normally distributed with mean μ and

standard deviation σ, the sampling distribution of is

also normally distributed with

• Z-value for the sampling distribution of is calculated:

X

μμX

n

σσ

X

X

n

σμ)X(

σ

)μX(Z

X

X

where: = sample mean

= population mean

= population standard deviation n = sample size

Xμσ

• If population is not normally distributed, we can apply the Central Limit Theorem which proves that:– …sample means from the population will be

approximately normal as long as the sample size is large enough.

• What is large enough?

• For most distributions, n > 30 will give a sampling distribution that is nearly normal

• For fairly symmetric distributions, n > 15

• For normal population distributions, the sampling distribution of the mean is always normally distributed

• (See page 238 for distributions of the population and samples)

Application:A brand name breakfast cereal company produces 5000 boxes

of serial per day. Each box is suppose to have 368 grams of cereal with an average dispersion of 15 grams.

• Set up the information in terms of population distribution.

• Questions:

1. What percent of individual boxes will have less than 365 grams?

2. If a sample of 25 boxes are selected what is the probability that the sample mean is less than 365?

3.If all possible samples of size 25 are taken, what interval around the population mean will contain 95% of all sample means?

4.What is the probability that a sample mean will be within the above estimated interval?

• All material in this chapter assumed sampling with replacement. Apply the Finite Population Correction (fpc) if:

– the sample is large relative to the population (n is greater than 5% of N)

and…

– Sampling is without replacement

• The fpc factor is 1N

nN