### Chapter 24: Comparing Means (when groups are

```Chapter 24: Comparing Means
(when groups are independent)
AP Statistics
Sampling Distribution for the Difference of Two
Means (when groups are independent)
Sampling Distribution for the Difference of Two
Means (when groups are independent)
Formula for degrees of
freedom when comparing
means of independent
groups
The calculator will compute
this for you
Assumptions and Conditions
Independence Assumption:
Randomization Condition
10% Condition
Normal Population Assumption:
Need to check each group for normality. SHOW GRAPH.
Nearly Normal Condition
Independent Groups Assumption
Just check for reasonability (this is very important)
Two-Sample t-interval
Two-Sample t-test
Example
Below are the saturated fat content (in grams)
for several pizzas sold by two national chains.
Create a 95% confidence interval for the
difference in the means for the saturated fat
content of the two brands.
Brand D
17 12 1 0 8 8 10 10 5 16 16 8 12 15 7 11 11 13 13 11 12
Brand PJ
6 7 11 9 4 4 7 9 11 3 4 5 8 5 5
Example
In order to create a twosample t-test, I first need
to satisfy the
Independent Sample
Assumption, the Normal
Population Assumption
and the Independent
Group Assumption. To
satisfy these, I will need
to satisfy the following
conditions
Example
To satisfy the Independent Samples Assumption,
we need to satisfy the below:
Randomization Condition: We can assume
that the pizzas from each company were
picked at random
10% Condition: We assume that the 20 and
15 pizzas are both less than 10% of the pizzas
made by each company
Example
To satisfy the Normal Population Condition, I can
satisfy the
Nearly Normal Condition (remember how sample
size plays a role in what we look for)
Brand D
Brand PJ
Both distributions of
saturated fat
roughly unimodal and
symmetric.
Example
To satisfy the Independent Groups Assumption, I
can assume that the groups are independent.
There is no reason to think that the fat
content in Brand D is not independent from
the fat content in Brand PJ.
Since all the Assumptions and Conditions have
been met, we can use a t-distribution with
32.757 degrees of freedom and create a twosample t-interval.
Example
nD  20
y D  11.25
nPJ  15 y D  6.53
y D  y D  4.72
sD  3.193
sPJ  2.588
df  32.757
Example

SE y D  y PJ

2
D
2
PJ
s
s


nD nPJ
2
2
3.193 2.588


 0.978
20
15
Example
y D  y PJ  t
*
32.8

SE y D  y PJ
4.72  2.030.978
4.72  1.99
2.73,6.71

Example
We are 95% confident that the true mean fat
content of Brand D is between 2.73 and 6.71
grams higher than the true mean fat content
for Brand PJ.
Example
Do the pizza chains have significantly different
mean saturated fat contents? Conduct a
hypothesis test.
```