Comparing 2
population
parameters
Chapter 13
Introduction: Two Sample
problems



Ex: How do small businesses that fail differ
from those that succeed? Business school
researchers compare the asset to liability
ratios of 2 samples of firms started in 2000: one
sample of failed businesses and one of firms
still going strong.
This observational study compares two
random sample, one from each of two
different populations.
Comparing two populations or two
treatments is one of the most common
situations in statistical practice.
13.1 We can examine two-sample data graphically
by comparing dotplots or stemplots (smaller samples)
or histograms or boxplots (larger samples). This
chapter details
Note! The difference between independent samples (this
chapter) and matched pairs or paired samples (past
chapters)
Conditions for comparing 2 means
*Note: If each population is normally distributed, then μ1 – μ2 will be too
Symbols
Parameters
Populaton Variable
Mean
Statistics
SD
Sample
Size
Mean
SD
1
X1
μ1
σ1
n1
x bar1
s1
2
X2
μ2
σ2
n2
x bar2
s2
There are four unknown parameters (the two means and
the two SD’s). We want to compare the two population
means, either by giving a confidence interval for their
difference (μ1 – μ2) or by testing the hypothesis of no
difference, H0: μ1 = μ2
Two Sample Z Statistic
 If
we know the population standard deviations for
both groups (unlikely) we use a 2 sample Z test. By
hand (formula below, same as before) or on calc
(2-SampZTest)
Two Sample T Test
 Degrees
of freedom for T*: Complicated to
calculate, so 2 options


1. Calc does it automatically! (decimals
possible
2. Compare the df of each sample (n1 – 1) and
(n2 – 2). Use the smaller of the 2 groups. This is
very conservative.
Robustness

The 2 sample t procedures are more robust than
the one-sample t methods

When planning a two sample study, choose equal
sample sizes if you can. When the shapes of the 2
populations are similar and the two samples are
equal sizes, p – values are quite accurate.

Two sample t procedures are most robust against
non-Normality and the conservative P-values
(using option 2 for df) are most accurate.
Example
 Does
increasing the amount of calcium in our diet
reduce blood pressure?


Double Blind, Randomized comparative experiment
with 21 healthy men. Group 1, chosen randomly, 10
men receiving a calcium supplement for 12 weeks.
Group 2, 11 men, received an identical placebo.
Response variable: The decrease in systolic blood
pressure for the subject after 12 weeks in MM of
Mercury (so a negative # indicates an increase
Group 1:
Group 2:
7, -4, 18, 17, -3, -5, 1, 10, 11, -1
-1, 12, -1, -3, 3, -5, 5, 2, -11, -1, -3
 1:

Hypothesis
H0 : μ 1 = μ 2
HA : μ 1 ≠ μ 2
or
H 0: μ1 - μ2 = 0
HA : μ 1 - μ 2 ≠ 0
2: Conditions
 SRS: Assume random selection of 21 subjects from
population of all men, and random assignment of
subjects to treatments confirmed in
 Normality: Small samples, check plots of each (no
serious non-normality or huge outliers)
Independence: “Because of the randomization, we are
willing to regard the calcium and placebo groups as two
independent samples. We are not sampling without
replacement from a population of interest in this case.
Group
Treatment
n
Xbar
1
calcium
10 5.000
S
8.743
2
placebo
11 -.273 5.901
3. Calculations:
We use the T test b/c we don’t know population sigma

5 - (-.0273)
2
8.7432 5.901
+
10
11
= 1.604
P value: There are 9 DF (the smaller of the 2 groups) which is
approximately .07
4: Interpretation
The experiment provides some evidence that
calcium reduces blood pressure, but the evidence
falls short of the traditional 5% and 1% levels. We
would fail to reject the null at either of these
significance levels.

90% Confidence interval: (t* is 1.833)
(-.753, 11.299). We are 90% confident that the true
mean advantage of calcium over a placebo lies in
this interval. Since 0 is in the interval, we cannot
reject the null.
*Remember, these are small samples! Bigger
samples (and bigger df) give smaller t* values
making them easier to ‘beat’ and attain
significance!
Calculator
 Enter
Group 1 data into L1 and Group 2 into L2
 Go to Stat/Tests choose 2-SampTTest
 Specify “data” Pooled- NO
 CI:
Choose 2-SampTInt then data, etc
 “Pooled”:
DON’T POOL!!!
13.2 Comparing 2 proportions
 Ex:
Does prayer help with in vitro
fertilization? Through random assignment,
88 women undergoing in vitro were
prayed for by intercessors and 81 were
not. 44 of the 88 women (50%) in the
treatment group got pregnant compared
to 21 out of the 81 (26%) in the control
group. This difference seems large, but is
it statistically significant?
Symbols
Population
population
proportion
Sample size
sample
proportion
1
ρ1
n1
ρhat 1
2
ρ2
n2
ρhat2
We compare the populations by doing
inference about the difference ρ1 - ρ2
Calculator
 Significance
 CI:
test: 2-PropZtest
2-PropZInt
Let’s do In Vitro Example

1: Hypothesis

H0: ρ1 = ρ2
H A : ρ1 ≠ ρ2

or
H0: ρ1 -ρ2 = 0
HA: ρ -ρ2 ≠ 0
2: Conditions
 SRS? Normality? Independence?
 3:
Calculations (on calc)
 4: Interpretation

“This study shows that intercessory prayer may cause
an increase in pregnancy. However, it is unclear if the
women knew that they were in a treatment group. If
they found out that other people were praying for
them, then their behaviors may have changed.
Continued
 CI
on calc + interpretation
 Explain
Type I and Type II error in this
setting. Which is more serious?