Embed Size (px)
Citation preview
SECTION 11.2COMPARING TWO MEANSAP Statistics
Comparing Two Means
AP Statistics, Section 11.2
2
Very useful compare two populations Two population equates to two
distributions, perhaps of different size Easier math to work with one distribution Distribution of the difference of means
and one sample t-procedures when possible.
Some formulas:3
AP Statistics, Section 11.2
z test
xz
n
t test
xz
s
n
Formulas (continued)
AP Statistics, Section 11.2
4
1 2
2 2 2 21 2 1 2
x x1 2 1 2
s s s s
n nn n
x
s
n n
2 2X Y X Y
x y x y
AP Statistics, Section 11.25
1 2 1 2
1 2
2 21 2
1 2
1 2
2 21 2
1 2
If then 0
x xz
n n
x xt
s s
n n
1 2 1 2
2 21 2
1 2
1 2 1 2
2 21 2
1 2
x xz
n n
x xt
s s
n n
Assumptions for Comparing Two Means
AP Statistics, Section 11.2
6
We have two SRSs, from two distinct populations.
The samples are independent. That is, one sample has no influence on the other. Matching violates independence, for example.
We measure the same variable for both samples.
Both populations are normally distributed. The means and standard deviations of the populations are unknown.
Example
AP Statistics, Section 11.2
7
Does increasing the amount of calcium in our diet reduce blood pressure? Examination of a large sample of people revealed a relationship between calcium intake and blood pressure. The relationship was strongest for black men. Such observational studies do not establish causation. Researchers therefore designed a randomized comparative experiment.
Example
AP Statistics, Section 11.2
8
The subjects in part of the experiment were 21 healthy black men. A randomly chosen group of 10 of the men received a calcium supplement for 12 weeks.
The group of 11 men received a placebo pill that looked identical.
The experiment was double-blind. The response variable is the decrease in systolic
(heart contracted) blood pressure for a subject after 12 weeks, in millimeters of mercury. An increase appears as a negative response.
Example
AP Statistics, Section 11.2
9
Group 1 (Calcium) results: 7, -4, 18, 17, -3, -5, 1, 10, 11, -2 n=10, x-bar=5.000, s=8.743
Group 2 (Placebo) results: -1, 12, -1, -3, 3, -5, 5, 2, -11, -1, -3 n=11, x-bar=-0.273, s=5.901
Inference Tool Box
AP Statistics, Section 11.2
10
Step 1: Identify the populations and the parameters of interest you want to draw conclusions about. State hypothesis in words and symbols. Pop1: Black Men on Calcium; Pop2: Black Men on Placebo Parameters of interest: mean differences in
blood pressure H0: µ1= µ2 (There is no difference in the blood
pressure changes) Ha: µ1> µ2 (The men taking calcium see a larger
decrease in blood pressure)
Inference Tool Box
AP Statistics, Section 11.2
11
Step 2: Choose the appropriate inference procedure, and verify the conditions for using the selected procedure. Test? Because we don’t know the population
standard deviation, we’ll use a t test. Since we’re not comparing a person with himself, we have two sample.
Independent? SRSs, therefore independent. Normal? Use back-to-back stemplots to check
for normality. The book says “no departures from normality”
Inference Tool Box
AP Statistics, Section 11.2
12
Step 3: Compute the test statistic and and the P-value.
2 2
5.000 ( .273)t 1.604
8.743 5.90210 11
p value :P(t 1.604) 0.0644
Notes on p-value
There are two options for calculating p-value:
Option 1: Use 2 sample t procedures from data and allow calculator to compute.
Option 2: Use procedures based on t-distribution with the smaller n to find d.f.
AP Statistics, Section 11.2
13
Exercises
AP Statistics, Section 11.2
14
11.2 HOMEWORK 11.33-11.35 all 11.37 – 11.39, 40, 43, 53, 54, 58, 62, 64 Due Friday, March27 Post Test Chapters 1 - 11 on Wed., April
1st
