chapter 24: comparing means (when groups are independent)
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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.
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