AP STATISTICS
LESSON 11 – 1
(DAY 3)
Matched Pairs t Procedures
ESSENTIAL QUESTION:
How are matched pair designs included
in one-sample data analysis?
Objectives:

To construct t confidence intervals for matched pair
designs.

To construct t significance tests for matched pair
designs.

To define Robustness of t procedures.
Matched Pairs t Procedures
The taste test in Example 11.3 was a matched pairs
study in which the same 10 tasters rated before-andafter sweetness. Comparative studies are more
convincing than single-sample investigations. For that
reason, one-sample inference is less common than
comparative inference.
One common design to compare two treatments makes
use of one-sample procedures. In matched pairs
design, subjects are matched in pairs and each
treatment is given to one subject in each pair.
Matched Pairs t Procedures
(continued…)
To compare the responses to the two treatments in a
matched pairs design, apply the one-sample t
procedures to the observed differences.
The parameter μ in a matched pairs t procedure is the
mean difference in the responses to the two treatments
within matched pairs of subjects in the entire
population.
Example 11.4
Page 629
Floral Scents and Learning



The randomization is important because subjects tend to
improve their times as they work a maze repeatedly.
To analyze these data, subtract the scented time from the
unscented time for each subject. The 21 differences form
a single sample.
Inference procedures for comparing two samples assume
that the samples are selected independently of each other.
This assumption does not hold when the same subjects
are measured twice. The proper analysis depends on the
design used to produce the data.
Example 11.5 Page 632
Is Caffeine Dependence Real?
Construct a 90% confidence interval for the mean change in
depression score.
We are 90% confident that the actual mean difference in depression
score for the population is between 3.579 and 11.149 points. We
estimated that caffeine-dependent individuals would score, on average,
between 3.6 and 11.1 points higher on the Beck Depression Inventory
when the are given placebo instead of caffeine.
This study provides evidence that withholding caffeine from caffeinedependent individuals may lead to depression.
Robustness of t Procedures
The t confidence interval and test are exactly correct
when the distribution of the population is exactly
normal. No real data are exactly normal.
Because of the tails of normal curves drop off quickly,
samples from normal distributions will have very few
outliers . Outliers suggest that your data are not a
sample from a normal population. Like x and s, the t
procedures are strongly influence by outliers.
Robustness Procedures
(continued…)
A confidence interval of significance test is
called robust if the confidence level or P-value
does not change very much when the
assumptions of the procedure are violated.
Example 11.6 Page 635
The Effects of Outliers
One of the 46 engines in our sample emitted an
unusually high amount (2.94 grams/mile) of
NOX. If remove that single data point, What
will happen.
 Fortunately, the t procedures are quite robust
against nonnormality of the population when
there are no outliers, especially when the
distribution is roughly symmetric.
The Effects of Outliers (continued…)



Large samples improve the accuracy of Pvalues and critical values from the t distribution
when the population is not normal. The main
reason for this is the central limit theorem.
Always make a plot to check for skewness and
outliers before you use the one-sample t
procedures when
N ≥ unless an outlier or quite strong skewness
is present.
Using the t procedure




Except in the case of small samples, the assumption
that the data are an SRS from the population of interest
is more important than the assumption that the
population distribution is normal.
Sample size at least 15. Use t procedure if the data
are close to normal. If the data are clearly nonnormal
or if outliers are present, do not use t.
Sample size at least 15. The t procedure can be used
except in the presence of outliers or strong skewness.
Large samples. The t procedures can be used even
for clearly skewed distributions when the sample is
large, roughly n ≥ 40.
Can we use t ?
Example 11.7
Page 636 and 637
 Figure 11.9(a) is a histogram of the percent of each
state’s residents who are over 65.We have data on the
entire population, we can find μ and σ so we do not
confidence intervals or tests.
 Figure 11.9 (b) Shows the time lightning strike each day
in the mountains of Colorado. The data has 70
observations that have a symmetric distribution. You can
use t procedures with confidence.
 Figure 11.9 (c) shows the distribution of word length in
Shakespeare’s plays is skewed to right. You can use the
t procedure for a distribution like this if the sample size is
roughly 40 or larger.
The power of a statistical test measures its
of from
the the
t test
ability toThe
detectpower
deviations
null
hypothesis. In practice we carry out the test in
the hope of showing that the null hypothesis is
false, so higher power is important. The power
of the one-sample t test against a specific
alternative value of the population mean μ is
the probability that the test will reject the null
hypothesis when the mean has this alternative
value. To calculate the power, we assume a
fixed level of significance, usually α = 0.05
Example 11.8
Page 639
Power of Spanish Teachers.