AP Stat Chapter 23 Blue Sheet
Comparing Means
Resting pulse rates for a random sample of 26 smokers had a mean of 80 beats per minute (bpm) and a
standard deviation of 5 bpm. Among 32 randomly chosen nonsmokers, the mean and standard deviation
were 74 and 6 bpm. Both sets of data were roughly symmetric and had no outliers. Is there evidence of
a difference in mean pulse rates between smokers and non-smokers?
Assumptions for Inference
1. Groups are independent
2. Data in each group are independent.
3. Both populations are Normal.
Condition That Support or Override Them
1. (Think about the design.)
2. SRS or random allocation, n < 10% of population
3. Both graphs show. . .
OR . . .
“The conditions are satisfied, so proceed with a t-model with _________ df, and conduct a two-sample ttest. “
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Chapter 24 Blue Sheet
Comparing Means
One indicator of physical fitness is resting pulse rate. Ten men volunteered to test an exercise device
advertised on television by using it three times a week for 20 minutes. Their resting pulse rates (bpm)
were measured before the test began, and then again after six weeks. Results are shown in the table.
Independence Assumption. If the data are paired, the groups are not
independent. It’s the differences that must be independent of each
other.
Randomization Condition: Pairs may be a random sample. Pairs may
be randomly assigned, or treatments may be randomly assigned to one
member of each pair.
10% Condition: Want the pairs to be no more than 10% of that
population if sampling from finite population
Nearly Normal Condition: check distribution of differences for
strong skew or outliers, or invoke CLT all hail if number of pairs > 30.
The paired t-test:
When conditions are met, test whether the mean of paired differences is significantly different from
zero. Test the hypothesis H 0 : µd = Δ 0 , where d’s are the pairwise differences and Δ 0 is almost always 0.
d − Δ0
Use the statistic, t n−1 =
, where d is the mean of the pairwise differences, n is the number of pairs,
SE ( d )
s
and SE ( d ) = d .
n
Paired t-interval
When the conditions are met, we are ready to find the confidence interval for the mean of the paired
differences. The confidence interval is d ± t * n−1 × SE ( d )