Download Binomial Proportions You have a random sample with frequencies

Welcome to .
Week 08 Thurs .
MAT135 Statistics
Estimation
We use the sample mean 𝒙 to
estimate the unknown population
mean µ
Estimation
Using the sample mean 𝒙 to
estimate the unknown population
mean µ is called “making
inferences”
Estimation
The sample standard deviation
“s” is the best estimate we have
for the unknown population
standard deviation “σ”
Estimation
Using s to estimate σ
is also
an inference
Estimation
s IS NOT the measure of
variability in the new population
of 𝒙s
Estimation
It needs to be decreased to take
sample size into account!
Estimation
We use:
s/ n
for the measure of variability
in the new population of 𝒙s
Estimation
The standard deviation
of the 𝒙s: s/ n
is called the “standard error”
abbreviated “se”
Estimation
So our curve is:
𝒙-3se 𝒙-2se 𝒙-se 𝒙
𝒙+se 𝒙+2se 𝒙+3se
Binomial Proportions
All of these activities can be
done when you have measured
data (continuous)
Binomial Proportions
What if you had counts in two
categories and wanted to
estimate the population
proportion?
Binomial Proportions
Weep?
Throw things?
Give up?
Binomial Proportions
Naw… we’ve got an easy solution!
http://www.shutterstock.com/s/whew/search.html
Binomial Proportions
Suppose you have a random
sample of size “n”
Binomial Proportions
Suppose you have a random
sample of size “n”
Suppose you want the
proportion of successes p
Binomial Proportions
Suppose you have a random
sample of size “n”
Suppose you know the number
of successes in your sample is
“x”
Binomial Proportions
The sample proportion
(symbolized: p, called: “p-hat”)
is:
x
p =
n
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 1
You have a random sample with
frequencies:
Blue
35
Is it binomial?
Red
23
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 1
You have a random sample with
frequencies:
Blue
35
Red
23
Is it binomial? yep
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 2
You have a random sample with
frequencies:
Blue
35
What is n?
Red
23
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 2
You have a random sample with
frequencies:
Blue
35
What is n?
58
Red
23
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 3
You have a random sample with
frequencies:
Blue
35
What is xred?
Red
23
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 3
You have a random sample with
frequencies:
Blue
35
What is xred? 23
Red
23
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 4
You have a random sample with
frequencies:
Blue
35
What is pred?
Red
23
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 4
You have a random sample with
frequencies:
Blue
35
Red
23
x 23
What is pred? =
≈ 40.0%
n
58
Questions?
Binomial Estimation
Just like the sample mean 𝒙 is
the best estimate of the true
population mean μ…
Binomial Estimation
…there is a true population
proportion “p” that is best
estimated by the sample
proportion p
Binomial Estimation
In symbols:
µp = p
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 5
You have a random sample with
frequencies:
Blue
35
Red
23
What is the best estimate for
pred?
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 5
You have a random sample with
frequencies:
Blue
35
Red
23
What is the best estimate for
pred? 40.0%
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 6
You have a random sample with
frequencies:
Blue
35
Red
23
What is the best estimate for
qred?
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 6
You have a random sample with
frequencies:
Blue
35
Red
23
What is the best estimate for
qred? 60.0%
Binomial Estimation
Because p-values distributions
are often skewed, you need to
be sure your sample size is
large before assigning
probabilities to this estimate
using a normal distribution
Binomial Estimation
You can assume normality if:
np ≥ 5or10
and n(1-p) ≥ 5or10
(nq ≥ 5or10)
Binomial Estimation
This is known as the
Rule of Sample Proportions
Binomial Estimation
Note: your book says the shape
of the sampling distribution of
p-hat is approximately normal
provided np(1-p)≥10
(I think it’s a typo)
Binomial Sampling
What if your sample size is too
small?
Binomial Sampling
The normal approximation can
always be used, but if the
conditions are not met, then
the approximation may not be
that good of an approximation.
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with
frequencies:
Blue
35
Red
23
Is pred likely to be normallydistributed?
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with
frequencies:
Blue
35
Red
23
Is pred likely to be normallydistributed? Is np ≥ 5or10?
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with
frequencies:
Blue
35
Red
23
Is pred likely to be normallydistributed? Is np ≥ 5or10?
np ≈ 58 × .400 ≈ 23.2
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with
frequencies:
Blue
35
Red
23
Is pred likely to be normallydistributed? Is nq ≥ 5or10?
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with
frequencies:
Blue
35
Red
23
Is pred likely to be normallydistributed? Is nq ≥ 5or10?
nq ≈ 58 ×.400
≈ 23.2
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with
frequencies:
Blue
35
Red
23
Is pred likely to be normallydistributed?
BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with
frequencies:
Blue
35
Red
23
Is pred likely to be normallydistributed? yep, close enough
Questions?
Binomial Sampling
Just as a gazillion samples each
with its own mean gives a new
population: the gazillion means
Binomial Sampling
…a gazillion samples each with
its own proportion gives a new
population: the gazillion ps
Binomial Sampling
And, again if you plotted the
frequency of the gazillion p
values, it would be called a
SAMPLING DISTRIBUTION
Binomial Sampling
And, again, the shape of the
plot of the gazillion sample
means would have a normal-ish
distribution
NO MATTER WHAT THE
ORIGINAL DATA LOOKED
LIKE
Binomial Sampling
Very nonnormal
population
Binomial Sampling
Very nonnormal
population
Normal-er
Binomial Sampling
Very nonnormal
population
Normal-ish
Binomial Sampling
And, again,
as “n”
increases, the
variability
(spread)
decreases
Binomial Sampling
The standard deviation of the ps
is:
σp =
p(1−p)
=
n
pq
n
Binomial Sampling
The standard deviation of the ps
is:
σp =
p(1−p)
=
n
pq
n
also called the standard error of
p-hat: sep
Binomial Sampling
So a normal curve would be:
p-3
pq
n
p-2
pq
n
p-
pq
n
p
p+
pq
n
p+2
pq
n
p+3
pq
n
Binomial Sampling
So a normal curve would be:
p-3 sep
p-2 sep
p-sep
p
p+sep
p+2 sep
p+3 sep
BINOMIAL SAMPLING
IN-CLASS PROBLEM 8
You have a random sample with
frequencies:
Blue
35
Red
23
What is the standard error of
p-hat?
BINOMIAL SAMPLING
IN-CLASS PROBLEM 8
You have a random sample with
frequencies:
Blue
35
Red
23
What is the standard error of
p-hat? ×
sep =
pq
=
n
.6×.4
≈ .064
58
BINOMIAL SAMPLING
IN-CLASS PROBLEM 9
You have a random sample with
frequencies:
Blue
35
Red
23
What would the normal curve
look like?
BINOMIAL SAMPLING
IN-CLASS PROBLEM 9
p-3 sep
p-2 sep
p-sep
p
p+sep
p+2 sep
p+3 sep
BINOMIAL SAMPLING
IN-CLASS PROBLEM 9
.6-3(.064) .6-2(.064)
.6-.064
.6
.6+.064
.6+2(.064) .6+3(.064)
BINOMIAL SAMPLING
IN-CLASS PROBLEM 9
.408
.472
.536
.6
.664
.728
.792
BINOMIAL SAMPLING
IN-CLASS PROBLEM 10
.408
.472
.536
.6
.664
.728
What is P(0.536<p<.664)?
.792
BINOMIAL SAMPLING
IN-CLASS PROBLEM 10
.408
.472
.536
.6
.664
.728
What is P(0.536<p<.664)?
68%
.792
BINOMIAL SAMPLING
IN-CLASS PROBLEM 11
.408
.472
.536
.6
.664
.728
.792
What is the range of values for
p that would with 95% certainty
include the true p?
BINOMIAL SAMPLING
IN-CLASS PROBLEM 11
.408
.472
.536
.6
.664
.728
.792
What is the range of values for
p that would with 95% certainty
include the true p? .472<p<.728
BINOMIAL SAMPLING
IN-CLASS PROBLEM 12
.408
.472
.536
.6
.664
.728
.792
What is the probability the true
p lies between 0.47 and .65?
BINOMIAL SAMPLING
IN-CLASS PROBLEM 12
.408
.472
.536
.6
.664
.728
.792
What is the probability the true
p lies between 0.47 and .65?
76.2%
Binomial Sampling
A study I did for my science
class…
Binomial Sampling
Some diamonds fluoresce under
UV light
Binomial Sampling
What percentage of diamonds
fluoresce?
Binomial Sampling
I inherited a bunch of jewelry
Some of it has diamonds in it
Binomial Sampling
Is this a binomial variable?
Binomial Sampling
What is n?
Fluoresce
8
Don’t
114
Binomial Sampling
What is x?
Fluoresce
8
Don’t
114
Binomial Sampling
What is your best estimate of p?
Fluoresce
8
Don’t
114
Binomial Sampling
Is it normal?
Fluoresce
8
Don’t
114
Binomial Sampling
To find out what sample size
you need to:
Estimate p
Use the min of p and q
Algebra magic:
n > 5or10/min(p,q)
Binomial Sampling
Fluoresce
8
Don’t
114
p ≈ .066 so q ≈ .934
so we’ll use p
n > 10/.066 or 151.5
Binomial Sampling
Fluoresce
8
Don’t
114
p ≈ .066 so q ≈ .934
so we’ll use p
n > 10/.066 or 151.5
In other words n > 152
Questions?
You survived!
Turn in your homework!
Don’t forget
your homework
due next week!
Have a great
rest of the week!
www.playbuzz.com