Comparing Means



Comparing two means is not very different
from comparing two proportions.
This time the parameter of interest is the
difference between the two means, 1 – 2.
So, the standard deviation of the difference
between two sample means is
 12  22
SD  y1  y2  

n1

n2
Since we don’t know the true standard
deviations, we need to use the standard errors
s12 s22
SE  y1  y2  

n1 n2


Independence Assumption (Each condition needs to
be checked for both groups.):
◦ Randomization Condition:
◦ 10% Condition:
◦ Nearly Normal Condition:
Independent Groups Assumption
When the conditions are met, we are ready to find the
confidence interval for the difference between means
of two independent groups:
 y1  y2   t

df
 SE  y1  y2 
where the standard error of the difference of the
means is
s12 s22
SE  y1  y2  

n1 n2
The critical value depends on the particular confidence level, C,
that you specify and on the number of degrees of freedom, which
we get from the sample sizes and a special formula.

The special formula for the degrees of
freedom for our t critical value is:
2
 s12 s22 
  
 n1 n2 
df 
2
2
1  s12 
1  s22 
  
 
n1  1  n1  n2  1  n2 

Because of this, we will let technology
calculate degrees of freedom for us!


A researcher wanted to see whether there is a
significant difference in resting pulse rates for
men and women. The data she collected are
shown below:
Male
Female
Count
28
24
Mean
72.75
72.625
Median
73
73
StdDev
5.37225
7.69987
Range
20
29
IQR
9
12.5
Create a 90% confidence interval for the
difference in mean pulse rates.

When the conditions are met, the standardized
sample difference between the means of two
independent groups
y1  y2    1  2 

t
SE  y1  y2 
can be modeled by a Student’s t-model with a
number of degrees of freedom found with a special
formula.



The hypothesis test we use is the 2-sample t-test for the
difference between means.
The conditions for the two-sample t-test for the difference
between the means of two independent groups are the
same as for the two-sample t-interval.
We test the hypothesis H0: 1 – 2 = 0, where the
hypothesized difference, 0, is almost always 0, using the
statistic
y  y 
t

1
2
0
SE  y1  y2 
When the conditions are met and the null hypothesis is
true, this statistic can be closely modeled by a Student’s tmodel with a number of degrees of freedom given by a
special formula. We use that model to obtain a P-value.


Someone claims that the generic batteries
seem to have last longer than the brand-name
batteries. To prove this claim, we measured the
lifetimes of 6 sets of generic and 6 sets of
brand-name AA batteries from a randomized
experiment.
Generic
Brand-name
Count
6
6
Mean
206 min
187.4 min
StdDev
10.3 min
14.6 min
Does this prove the claim?


Page 640 – 643
Problem # 1, 3ab, 11, 13, 15, 17, 25, 31.