MATH 243, LECTURE 19
1. One-sample t-statistics
We are working in our basic framework: there is some measurement we’d like to understand over a large
population; sometimes there is some guess about that measurement we would like to test; so we take great
pains to obtain a random sample; the question is then “what can we deduce about the entire population
from just knowing this sample?” This is the basic question of study in statistics.
Suppose the measurement we are interested is some mean and we take a sample of size n. In one-sample
t-statistics we compute the sample mean - call it ¯x. We compute the deviation of the sample, which is
called the standard error - call it s. We use ¯xand sto compute confidence intervals or test hypothesis in
pretty much the same manner as we did before with splaying the role of σand, most importantly, using
the appropriate t-distribution (for size n) to convert between deviations and probabilities.
•With probability C% the (true) mean µhas values between x−t∗σ
√nand x+t∗σ
√n, where t∗is
the critical value associated to the t(n−1)-distribution such that there is a C% chance of finding
a value between −t∗and t∗.
•To test the null hypothesis, H0:µ=µ0, we calculate
t0=x−µ0
s/√n
Then P(t≥ |t0|)< α supports the alternative hypothesis Ha:µ6=µ0at level α.
Example 1. Use t-statistics to say whether a sample of size n= 81 with x= 52.1and s= 3.1can be used
to reject a null-hypothesis of µ≤50 at the 5% and 1% levels.
We can translate other kinds of problems, such as finding sufficient sample sizes, immediately as well.
Example 2. With data as above, how big a sample size is needed to achieve a margin of error of 0.2with
a confidence level of 99%?
1.1. Matched pairs t-procedures. As we saw earlier in the quarter, experiments are more convincing
than surveys, so it is helpful to see the use of t-statistics in this setting. In a matched-pairs experiment,
similar subjects are matched up and then at random given two different treatments, with the results then
measured and compared.
Example 3. Suppose a matched-pairs experiment is set up comparing two diets: the SeeFood diet and the
HiPhat diet. Matched pairs each go on one of these diets and the difference in weight loss is taken in each
pair. The data over 18 matched pairs is as follows:
3.5,−6.7,9.5,−7.3,0.5,5.0,5.0,5.3,−6.1,6.8,
3.8,−2.2,−1.3,1.7,−1.6,−3.1,6.2,9.5
(Here a value of 3.5 means that in the first pair the SeeFood person “lost” 3.5 pounds more than the HiPhat
person (it could be that the SeeFood person gained 5 pounds while the HiPhat person gained 8.5 pounds)).
Test the hypothesis that SeeFood leads to more weight loss than HiPhat at level α= 5%. (To get started:
x= 1.58.s= 5.32.)
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