Transcript Statistical Analysis – Chapter 5 “Central Limit Theorem”

Statistical Analysis – Chapter 5
“Central Limit Theorem”
Dr. Roderick Graham
Fashion Institute of Technology
The Central Limit Theorem

In chapter 4, we looked at drawing samples from a
BINOMIAL POPULATION

Now, we are concerned with predicting means for
sampling distributions (not individual values)

We will call the distributions of these samples “x-bar
distributions ” X
Equations for x-bar distributions….
(when we want to know information about our
sample drawn from a population)
X  
Mean of X-bar
distribution
 X  / n
Standard Deviation of X-bar
distribution
(in SPSS called “Standard Error”)
z
X 

X
Z- score (in this case, you
must use standard deviation
of X-Bar)
With these equations, you can use the normal table to predict SAMPLE
DISTRIBUTION VALUES.
The large histogram is the
distribution of individual values.
The small x-bars are distributions of
sample averages for the individual
values
If we collect enough sample
averages, the averages will be
normally distributed, and look like a
normal curve
Because of this, distribution, we can
make predictions
Taking Samples from Populations
This is Parent’s Age from the data we used for our
project. µ = 44.74. Let’s imagine that this is the
true population value. (The N is large, so we can
assume this.)
Taking Samples from Populations
Because the true population is normal, any sample
we draw will have approx. the same mean as the
true population, and a stand. dev. that decreases as
the N of our sample increases.
Taking Samples from Populations
Sample N = 92
 X = 44.86
 X  .715
Sample N = 266
 X = 44.33
 X  .453
Pop. Mean = 44.74
Sample N = 502
 X = 44.46

X
 .326
Standard
deviation
changes as
N changes
Means of
sample
close to
means of
population
The Central Limit Theorem
For sample sizes with n > 30, the sample distribution (x-bar
distribution) will be normally distributed with mean µ and
standard deviation σ/√n.
Note: If we assume that the population is normally distributed,
the sample size is not important.
The Central Limit Theorem

Because of the central limit theorem, you can use the
normal table for any random sample over 30 respondents
to find the probability of sample averages!

The main idea behind the problems we will work today is
that we are trying to understand the probabilities of
getting sample values using the z scores and the normal
table.
When solving x-bar problems…
X  
 X  / n
z
X 

X
The mean of the distribution is the mean of
the true population
The n is the total number of respondents in
your sample.
You use this z – score to find percentages on
the normal table.
Rounding Technique for the Remainder
of the Course
1.
Work in three decimal places for the entire problem
2.
Round your final answers to two decimal places
3.
Z-scores are always presented in two decimal places
(because on the normal table chart, z-scores are up to
two decimal places)
Sample Problems…

Let’s do this problem (problem 5.1) I’ll do A, and then I’ll
get volunteers to help us set up B and C as a class.
A machine cuts pieces of silk material to an average
length of 1000 mm with a standard deviation of 12 mm.
Between what two lengths would we expect to find the
middle 95% of all sample averages?
n = 36
b) n = 144
c) n = 576
a)
Sample Problems
Answer to problem 5.1
1.
2.
3.
a.
b.
c.
Calculating the standard deviation for each N.
Find the z-score associated with 47.5%. This because we need 95%, and
we need 47.5% on the positive side 0, and 47.5% for the negative side of
0. The z-score for 47.5% is +/-1.96.
Then you use the z score formula to solve for X-bar.
Between 996.08 mm and 1003.92 mm
Between 998.04 mm and 1001.96 mm
Between 999.02 mm and 1000.98
THIS PROBLEM SHOWS THAT THE LARGER OUR SAMPLE SIZE, THE
MORE ACCURATE OUR PREDICTIONS BECOME!
Sample Problems….(5.8, p. 157)
A nationwide marketing study concluded that the average age of horror
film moviegoers is 17.4 years old with a standard deviation of 2.7 years.
a.
Assuming a normal distribution, what percentage of horror film moviegoers
nationwide would you expect to be over 18 years old? (hint: when we
assume a normal distribution, we don’t worry about the N of the sample)
b.
If we take random samples of 81 horror film moviegoers nationwide and
calculate the sample average for each sample, what percentage of sample
averages would you expect to be over 18 years? (hint: now we are not
assuming a random sample, and we are given an N)
Sample Problems…

Let’s do problem 5.12
A survey indicated the average yearly salary of entry-level
women managers to be µ = $56,700 with σ = $7,200.
a) Assuming a normal distribution, what is the probability a
woman manager’s entry level salary will exceed $58,000?
b) What is the probability a random sample of 50 women
managers will yield an average entry-level salary exceeding
$58,000?
d) Assuming n = 42, with what probability can we assert a
sample average will fall within $1500 of µ = $56700?