1. Paired Samples
Chapter 25 part 1

2. One indicator of physical fitness is resting pulse rate
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 in beats per minute were measured before the test began, and then again after six weeks. Results are shown in the table. Is there evidence that this kind of exercise can reduce resting pulse rates? How much?
Subject
Pulse Before (bpm)
Pulse After (bpm)
Allen 73
Brandon 83 79
Carlos 85 81
David 87 86
Edwin 91
Franco 99
Graeme 84
Hans
Ivan
Jorge 76

3. Because the samples are not independent, we can’t do a 2-sample t-test
Because the samples are not independent, we can’t do a 2-sample t-test. The conditions are not met.
Instead, we can do two 1-sample
t-tests for the difference.

4. The d stands for the difference.
Step 1: State the Hypotheses
𝐻 0 : 𝜇 𝑑 ≤0
𝐻 𝐴 : 𝜇 𝑑 >0
The d stands for the difference.

5. Step 2: Conditions/Model
Calculate the difference of the before/after pulse rates in L3. Highlight L3, hit ENTER, and type L1-L2

6. Step 2: Conditions/Model
Now look at the histogram for the differences by plotting L3. This is the histogram that should be nearly Normal.
Now do a matched pairs t-test.

7. Step 3: Mechanics (Using L3)
𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑖𝑟𝑠=10
𝑑𝑓=𝑛−1=9
𝑑 =𝑚𝑒𝑎𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠=2.8
𝑠 𝑑 =𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠=2.53
𝑡= =3.5
P-𝑣𝑎𝑙𝑢𝑒=𝑃 𝑡>3.5 =0.0034

8. Step 4: Conclusion
With a low P-value of , we reject the null hypothesis. There is evidence to support the claim that this form of exercise can reduce resting pulse rates.

9. Step 5: 95% Confidence Interval (because we were asked how much)
𝑑 =2.8
2.8± =(0.99, 4.61)
𝑡 9 ∗ =2.262
𝑆𝐸 𝑑 = =
We are 95% confident that the “before” pulse rate is between 0.99 and 4.61 beats higher than the “after” pulse rate, which indicates a decrease of this amount after six weeks of this exercise program.

10. Today’s Assignment:
HW: page 604 #16
Ch. 23 & 24 Homework due today
Test 5 is coming soon – after Spring Break