Download The Sampling Distribution of the Difference Between Two Sample

§10.1–The Sampling Distribution of the Difference
Between Two Sample Means for Independent Samples
Tom Lewis
Fall Term 2009
Tom Lewis ()
§10.1–The Sampling Distribution of the Difference Between Two
FallSample
Term 2009
Means for1 Independent
/6
Samples
Outline
1
The rationale
2
A small example
3
Normal populations
Tom Lewis ()
FallSample
Term 2009
/6
§10.1–The Sampling Distribution of the Difference Between Two
Means for2 Independent
Samples
The rationale
A typical problem
Do women do better on the SAT than men? How could we test for this?
There are two populations under consideration: the men and the
women.
There is a common statistic under consideration: the SAT score.
Each population has its own population mean SAT score: µ1 for the
boys and µ2 for the girls.
We can collect random samples from each population and compute
the sample means of their SAT scores: x 1 for the boys and x 2 for the
girls.
How can we compare the sample means? How much of a difference
between the sample means, x 2 − x 1 , is sufficient to assert that there
is a difference in the population means, µ2 − µ1 .
§10.1–The Sampling Distribution of the Difference Between Two
FallSample
Term 2009
Means for3 Independent
/6
Samples
Tom Lewis ()
A small example
A small example
Here are the scores on a recent exam for a group of boys and girls:
Alex
55
Bob
75
Chuck
68
Denise
82
Ellen
76
Fergie
88
Gisele
50
Find all samples of size 2 from the boys and all samples of size three
from the girls. Find the values of the mean of the scores for each of
the random samples. Let x 1 be the mean of the boy’s samples and let
x 2 denote the means of the girl’s samples.
Find all 12 possible values of x 1 − x 2 .
Find the mean and standard deviation of the values of x 1 − x 2 .
Tom Lewis ()
FallSample
Term 2009
/6
§10.1–The Sampling Distribution of the Difference Between Two
Means for4 Independent
Samples
Normal populations
Normal data
Our next result is significant, but it requires that the variable under
question be normally distributed within the two populations.
Theorem
Suppose that x is a normally distributed variable on each of two
populations. Then, for independent samples of size n1 and n2 from the
two populations,
µx 1 −x 2 = µ1 − µ2 ,
q
σx 1 −x 2 = (σ12 /n1 ) + (σ22 /n2 )
x 1 − x 2 is normally distributed.
Tom Lewis ()
§10.1–The Sampling Distribution of the Difference Between Two
FallSample
Term 2009
Means for5 Independent
/6
Samples
Normal populations
Problem
Work problems 10.10 and 10.18 from the text.
Tom Lewis ()
FallSample
Term 2009
/6
§10.1–The Sampling Distribution of the Difference Between Two
Means for6 Independent
Samples