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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