L07 Ch 07: Sampling and Sampling Distributions

Population

Census

Real

Parameter

Sample

Statistic

estimates

Statistical Inference

Parameters vs. Statistics

Numerical characteristics about a population

are called _________

◦ Mean, Standard Deviation, etc…

Numerical characteristics about a sample are

called ______________

◦ Mean, Standard Deviation, etc...

◦ Statistics ESTIMATE parameters

◦ X-bar is a point estimate of the population mean

It estimates the population mean with a single point rather

than an interval

◦ s is a point estimate of s

Population Parameter

Point Estimator

Point Estimate

Parameter Value

m = Population mean SAT score

990

s = Population std. deviation for SAT score

80 s = Sample std. deviation for SAT score

p = Population pro- portion wanting campus housing

.72

Summary of Point Estimates

Obtained from a Simple Random Sample

= Sample mean SAT score x

= Sample pro- portion wanting campus housing

p

Sampling Distributions

Turns out that sample statistics have their own shape!

That is right, they have their own distributions.

To see why: Consider the classroom a population of

interest. Want to know average height but collecting

census is too ―costly‖

◦ Form SAMPLES of size three

◦ In your sample, calculate the average height in inches.

◦ Notice that before a sample is drawn, x-bar is not known.

Therefore it is called a random variable

◦ Will all samples produce the same value for x-bar?

◦ How many unique samples of size 3 can I draw?

◦ Use formula for ______________

Visualize Sampling Distributions

In what follows, we will:

◦ Take a sample from a population of

observations

◦ From the sample, we will calculate the mean

◦ We will plot the mean on a number line

◦ We will repeat many, many times

◦ Goal: See the shape, or distribution of our

sample statistic x-bar

Link

Sampling Distribution of X-Bar

The sampling distribution (shape) of x-bar

has a center and it has spread.

The center of the x-bar distribution is

located at the mean of the underlying

__________

E( ) = mxbar = mx

Where m is mean of the population of

data points your are drawing from

x

Sampling Distribution of X-Bar

Standard Deviation of X-Bar, Two cases

1. Sampling from Finite Population

◦ Where N is the population size,

n is sample size and

is the finite population correction factor.

◦ When n—the sample size—is less than 5 percent

of N—the population sample size—then you

can ignore the correction factor.

2. Infinite population

Standard Deviation of a statistic is called

the standard error

◦ Above is the standard error of x-bar

1N

nN

nx

ss

1N

nN

x

x

What is the shape of X-Bar

Central Limit Theorem tells us the shape

of x-bar

If the sample size is ―big enough‖ then the

shape of x-bar is that of the ________

__________________

◦ It does not matter what the underlying

distribution is.

If the underlying distribution is normal, x-

bar is ____________no matter the

sample size.

Link

Link

Making Probability Statements about X-bar

A population has a sample mean of 200 and a standard

deviation of 50. Suppose a simple random sample of

size 100 is selected and x-bar is used to estimate m

◦ What is the distribution of x-bar?

◦ What is the probability that the sample mean will be within +

or – 5 of the population mean?

Notice how this builds on chapter 6!

We are going to use the formula to convert to z.

Instead of dividing by the standard deviation of x, we are

going to be dividing by the standard deviation of x-bar.

x

xZ

s

m

OLD Z

FORMULA

New Z

FORMULA

Proportions

Can we find the proportion of accounting majors in a

population?

Can we find the proportion of accounting majors in a

sample?

What would be a general name for these two

quantities?

If we take two samples, from the same population, are

we likely to get the same value for the proportion?

◦ ___________

◦ If we plot ____________, we get a shape of the proportion

statistic

The sample proportion is a ―statistic‖ and hence it has

its own shape called the ―sampling distribution‖

Proportions

P-bar = sample proportion

The shape can be approximated by the

___________________ when

◦ np ≥ 5 number in sample times proportion of

accounting majors is greater than 5

◦ and n(1-P) ≥ 5 number in sample times non

accounting majors

The center of 𝑝 is at p the population

proportion

E p p( )

Proportion

Standard Deviation of p-bar

n

ppp

)1( s𝝈𝒑 =

𝒑(𝟏 − 𝒑)

𝒏

𝑵 − 𝒏

𝑵 − 𝟏

Finite Population Infinite Population

• If sample is

______relative to

population

• If n/N > .05

• If sample is

______relative

to population

• If n/N < .05

Proportions

Why do we care about the shape of the

sample proportion?

◦ Helps us to understand how close or far away

we are from the POPULATION proportion.

A simple random sample of size 100 is

selected from a population with p = .40

◦ What is the expected value of 𝑝?

◦ What is the standard error of 𝑝?

◦ Show the distribution of 𝑝

◦ What does the sampling distribution of 𝑝 show?

Properties of Point Estimators Let’s talk about the desirable properties of

point estimators

◦ Wouldn’t it be nice if you expected your point

estimate to be a correct guess?

____________

This is why we use the finite correction factor and why we

divide by n-1 when we first introduced the sample standard

deviation.

◦ Wouldn’t it be nice if your estimate did not have a lot

of variation?

__________

◦ Wouldn’t it be nice if large sample sizes provided

better estimates?

__________

𝐸(𝜃) = 𝜃