Sampling Distributions

The “What If?” game

ParametersOValues that describe a

characteristic of the POPULATIONO Most of the time, there is no way for us to

really know what this number is

Oμ = meanOσ = standard deviationOp (or π) = proportionOα = y-int. of LSRLOβ = slope of LSRL

Of the POPULATIO

N

StatisticsOValue computed from a sampleO = meanOs = standard deviationO = proportionOa = y-int of LSRLOb = slope of LSRL

Of the SAMPLE

DistributionOAll the values a variable

can take, and the number of times that it takes each value

OA distribution is just a picture of the data

Put them together

Sampling Distribution

OThe distribution of possible values of a statistic, from all the possible samples of the same size, from the same population

Sampling Distribution

O I take a sample from a population, calculate a statistic.

O What if I could take every possible sample of that size from that population, and calculate the same statistic every time, and then plot all of these values

OThe picture (distribution) of all those statistics from all the samples (sampling)

Sampling Distributions

OWe are going to be concerned with the distributions of sample proportions, , and the distributions of sample means,

OSince we often don’t know the true proportion, p, or the true mean, μ, the only information we have to base decisions on is the statistics and

Sample Proportions

O = # in the sample that have this characteristic

Sample size

Assumptions - Proportions

O If we assume that our sample is not too big, less than 10% of the population so we can have independence

O And

O If we assumer our sample is big enough, where np > 10 and n(1 – p) > 10

OThen we can use a normal curve to approximate the sampling distribution

If we’re going to use a normal model, we

need:OMeanOStandard deviation

Suppose we have a population of six people: Alice, Ben, Charles, Denise, Edward, & Frank

We are interested in the proportion of females.

This is called the parameter of interest.

What is the proportion of females?

Draw samples of two from this population.

How many different samples are possible?

6C2 =15

Find the 15 different samples that are possible & find the sample proportion of the number of females in each sampleAlice & Ben .5Alice & Charles .5

Alice & Denise 1

Alice & Edward .5

Alice & Frank .5

Ben & Charles 0

Ben & Denise .5Ben & Edward 0

Ben & Frank 0Charles & Denise .5Charles & Edward 0Charles & Frank 0Denise & Edward .5Denise & Frank .5Edward & Frank 0

Find the mean & standard deviation of all p-hats.

29814.0σ&31

μ ˆˆ pp

Once you have your distribution of all the sample proportions in the whole wide world, from this size sample from the population…

n

pp

p

p

p

1σ

μ

ˆ

ˆ

These are found on the formula chart!

The mean of all the sample proportions (statistics) in the whole wide world, all the p-hats, is equal to the value of the proportion for the whole population (parameter)

Sample Means

O = Add all the individual values

Divide by how many there are

Assumptions - MeansOWe want to be able to use a

normal modelOCentral Limit Theorem – When

n is sufficiently large, the sampling distribution of is well approximated by a normal curve, even when the population distribution itself is not normal

Assumptions - Means

OSo, what is “sufficiently large”?

On ≥ 30

Consider the population of 5 fish in my pond – the length of fish (in inches):

2, 7, 10, 11, 14What is the mean and standard deviation of this population?

Let’s take samples of size 2

(n = 2) from this population:

How many samples of size 2 are possible?5C2 =

10Find all 10 of these samples and record the sample means.

What is the mean and standard deviation of the sample means?

mx = 8.8

sx = 2.4919

Repeat this procedure with sample size

n = 3Find all 10 of these samples and record the sample means.

What is the mean and standard deviation of the sample means?

mx = 8.8

sx = 1.6613

mx = m

sx =

The mean of all the sample means (statistics) in the whole wide world, all the x-bars, is equal to the value of the mean for the whole population (parameter)

These are found on the formula chart!