### Chapter 9

```Sampling
Distributions
Welcome to inference!!!!
Chapter 9
Parameter

A Parameter is a number that describes the population.


A parameter always exists but in practice we rarely know
it’s value b/c of the difficulty in creating a census.
We use Greek letters to describe them (likeμor σ). If we
are talking about a percentage parameter, we use rho (ρ)
Ex: If we wanted to compare the IQ’s of all American and
Asian males it would be impossible, but it’s important to
realize that μAmericans and μmales exist.
 Ex: If we were interested in whether there is a greater
percentage of women who eat broccoli than men, we
want to know whether ρwomen > ρmen

Statistic



A statistic is a number that describes a sample. The value of a
statistic can always be found when we take a sample . It’s
important to realize that a statistic can change from sample
to sample.
Statistics use variables like x bar, s, and phat (non greek).
We often use statistics to estimate an unknown parameter.



Ex: I take a random sample of 500 American males and find
their IQ’s. We find that x bar = 103.2.
I take a random sample of 200 women and find that 40 like
broccoli. Then phat w = .2
IMPORTANT! A POPULATION NEEDS TO BE AT LEAST 10 TIMES AS
BIG AS A SAMPLE TAKEN FROM IT. IF NOT, YOU NEED A
SMALLER SAMPLE
Bias
 We
say something is biased if it is a poor
predictor
Variability
*Variability of population doesn’t change- (scoop example)
size of scoop matters
How can we use samples to find parameters if
they give us different results?

Imagine an archer shooting many arrows at a target: 4
situations can occur

a) consistent but off target. b) all over the place. tends
to average a bulls eye but each result is far from center.
c) worse than a as the archer is consistently missing high
and to the right but nearly as consistently as situation a.
d) is ideal- low bias and low variability.
Here’s an example:


Suppose our goal was to estimate μamerican male.
We can’t take a census, so we take many samples. We
find the average IQ of american males in each sample (x
bar).

a) if our many samples of IQ’s are consistent but higher
than the true average IQ of AM’s, then we have a
situation with high bias and low variability

b) If our many samples are inconsistent- some high, some
low than the true mean of AM’s Iqs, we have low bias and
high variability.
 If
our sample means are not close to each
other but all higher than μAM then we
have high bias and high variability (c)
 Finally,
if our samples all just slightly higher
or slightly lower than μAM, we have our
desired situation: low bias and low
variability.
But…

You aren’t taking a bunch of samples…you’re only
going to take 1! and we want it to predict μam

If we used the data from situation d) then any of the
samples would provide a good predictor for μam

We already know some ways to get a good sampleusing an SRS and being very sure to have no bias
when choosing our sample.

Inference is using our sample statistic (assuming it’s a
good sample) to predict our parameter with a
certain degree of confidence.
The Sampling Distribution
 The
sampling distribution of a statistic is the
distribution of means of all possible samples of
the same size from the population.
 When we sample, we sample with replacement.
 A sampling distribution is a sample space- it
describes everything that can happen when we
sample.
Cool demo
Central Limit Theorem (CLT)
 As
you take more and more SRS’s of the same
size, the distribution of their means will get closer
and closer to a normal curve centered around
the true population mean…NO MATTER WHAT THE
SHAPE OF THE PARENT POPULATION!! Why?!
 The
Sampling Distribution of means has a mean of
μand a standard deviation of σ/√n**
N(μ, σ/√n)
CLT summary




1. The mean of the population (what we
want to find) will be the same as the mean of
all your many samples.
2. The Standard Deviation of all your many
samples will be the population standard
deviation divided by √n (your sample size)
3. The histogram of the samples will appear
normal (bell shaped).
4. The larger the sample size (n), the smaller
the standard deviation will be and the more
constricted the graph will be.
What if we’re talking about
proportions?
 Same
thing except we use our proportion
formulas for mean and standard
deviation
Don’t forget our rule of 10!
 Only
use σ/√n or
if the 10 times N (the number of people in
our population) is ≥ n (the number of
people in our sample)
Example

The true average study time for a final exam in
history is found to be 6 hours and 25 minutes with a
standard deviation of 1 hour and 45 minutes.
Assume the distribution is normal. N(6.417, 1.75)



What is the probability that a student chosen at
random spends more than 7 hours studying?
Normalcdf(7,100,6.417,1.75) = 37%
What is the probability that an SRS of 4 students will
average more than 7 hours in studying?
Normalcdf(7,100,6.417,1.75/√4) = 25.3%.
Why did the probability go down?

A student to study more than 7 hours is not probable…a
group of 4 to average more than 7 is less probable.
Example 2

The length of pregnancy from conception to birth
varies normally with a mean of 266 days and a
standard deviation of 16 days


N(266,16)
What is the probability that a woman chosen at
random has a pregnancy lasting more than 270
days?



40.1%
What is the mean and standard deviation of my
sampling distribution?
What is the probability that an SRS of 16 women
have pregnancies averaging more than 270 days?
```