Example: One-Sample T-Test
• Researchers are interested in whether the pulse
rate of long-distance runners differs from that of
other athletes
• They randomly sample 8 long-distance runners,
measure their resting pulse, and obtain the
following data:
45, 42, 64, 54, 58, 49, 48, 56
• The average resting pulse of athletes in the
general population is 60 beats per minute
• Test the null hypothesis at the 0.05 level of
significance
Example: One-Sample T-Test
HO: Pulse of long-distance runners = 60
HA: Pulse of long-distance runners differs from 60
•
•
•
•
•
•
Mean = 416/8 = 52
SS = 374; s2 = 53.4; s = 7.3; SE = 2.6
df = 7
t = (52-60)/2.6 = 3
tcrit (from table) at alpha of 0.05 = 2.365
Reject null hypothesis. There is a difference
Paired T-Test
• Used when two samples are not
independent of each other
• Observations in one sample can be paired
with observations in the other sample
• For example:
– Before and after observations on the same
subjects
– A comparison of two different measurements
or treatments on the same subjects
Paired T-Test: Procedure
• Calculate the difference (di = yi − xi) between the two
observations on each pair, making sure you distinguish
between positive and negative differences
• Calculate the mean difference
• Calculate the standard deviation of the differences (sd)
and use this to calculate the standard error of the mean
difference
• SE = sd / n
• Calculate t = d / SE
• Degrees of freedom = n − 1
• Use Table B.3 to obtain tcrit
Example
• Four individuals with high levels of cholesterol went on a
special diet, avoiding high cholesterol foods and taking
special supplements. Using the .05 level of significance,
was there a significant decrease in cholesterol level?
• Their total cholesterol levels before and after the diet
were as follows:
Before After
287
255
305
269
243
245
309
247
Example
Before
287
305
245
309
After
255
269
243
247
Difference
32
36
2
62
132 (Total)
Mean difference = 132/4 = 33
Standard deviation = 1812/3 = 24.6
Standard error = 12.3
t = 33/12.3 = 2.683
df = 3
tcrit = 2.353 (one-sided test at α = 0.05)
Reject null hypothesis
d-d
1
3
31
29
(d-d)2
1
9
961
841
1812 (SS)
CI for Difference in Means
• Recall example of verbal skills in 8-year boys and girls
C. I. =
X1 – X2  tcrit (SX1 – X2)
• tcrit = 2.878 for α =0.01
• 99% CI = 37-31 ± 2.878(2) = 6 ± 5.756 = (0.244; 11.756)
• This interval does not include zero, therefore there is a
difference between boys and girls