The “What If?” game

O Values that describe a characteristic of the POPULATION
O
O
Most of the time, there is no way for us to really know what this number is
μ = mean Of the
O
O p (or π ) = proportion
O α = y-int. of LSRL
O β = slope of LSRL

O Value computed from a sample
O
O
Of the 𝑥 = mean
SAMPLE s = standard deviation
O 𝑝 = proportion
O a = y-int of LSRL
O b = slope of LSRL

O All the values a variable can take, and the number of times that it takes each value
O A distribution is just a picture of the data

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

O
O
O
I take a sample from a population, calculate a statistic.
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
The picture (distribution) of all those statistics from all the samples
(sampling)

Sampling Distributions
O We are going to be concerned with the distributions of sample proportions, 𝑝 , and the distributions of sample means, 𝑥.
O Since 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 𝑝 and 𝑥.

O 𝑝 = 𝑥 𝑛
# in the sample that have this characteristic
Sample size

Assumptions - Proportions
O
O If we assume that our sample is not too big, less than 10% of the population so we can have independence
And
O If we assumer our sample is big enough, where np > 10 and n(1 – p) > 10
O Then we can use a normal curve to approximate the sampling distribution

If we’re going to use a normal model, we need:
O
O

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?
6
C
2
=15

Find the 15 different samples that are possible & find the sample proportion of the number of females in each sample
Alice & Ben .5
Alice & Charles .5
Alice & Denise 1
Alice & Edward .5
Alice & Frank .5
Ben & Charles 0
Ben & Denise .5
Ben & Edward 0
Ben & Frank 0
Charles & Denise .5
Charles & Edward 0
Charles & Frank 0
Denise & Edward .5
Denise & Frank .5
Edward & Frank 0
Find the mean & standard deviation of all p-hats.
μ 
1
3
& σ  0 .
29814

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

 p
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)
 p

 p
 n
These are found on the formula chart!

O
𝑥 =
𝑥 𝑖 𝑛
Add all the individual values
Divide by how many there are

O We want to be able to use a normal model
O Central Limit Theorem – When n is sufficiently large, the sampling distribution of 𝒙 is well approximated by a normal curve, even when the population distribution itself is not normal

O So, what is
“sufficiently large”?
O n ≥ 30

Consider the population of 5 fish in my pond – the length of fish (in inches):
2, 7, 10, 11, 14
What 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?
5
C
2
= 10
Find all 10 of these m x
What is the mean
= 8.8
s x
= 2.4919
means .
sample means?

Repeat this procedure with sample size n = 3
Find all 10 of these samples and record the sample means .
What is the mean and standard deviation of the sample means?
m x
= 8.8
s x
= 1.6613

m x
= m s x
= 𝜎 𝑛
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!