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 Condition10% Condition

Normal Population Assumption:Need to check each group for normality. SHOW GRAPH.

Nearly Normal Condition

Independent Groups AssumptionJust 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 two-sample 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 fatroughly 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 two-sample t-interval.

Example

757.32 72.4

588.2 53.6 15

193.3 25.11 20

dfyy

syn

syn

DD

PJDPJ

DDD

Example

978.015

588.2

20

3.193

22

22

PJ

PJ

D

DPJD n

s

n

syySE

Example

71.6,73.2

99.172.4

978.003.272.4

*8.32

PJDPJD yySEtyy

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.