Chapter 9. Comparing Two populations.
Section 9.1 and 9.2. Comparing Means.
Example 1. A biologist who studies spiders is interested in comparing the lengths of
female and male green lynx spiders. Assume that the length X of the male spider is
N(X,2X). You have a sample of size n from this population. Assume that the length Y
of the female spider is N(Y,2Y). You have a sample of size m from this population.
Assume that the X data is independent of the Y data.
Male lengths: 6.3, 6.3, 4.6, 6.5, 5.8 (m=5, x_bar =5.9, sX=0.75) 4 6
Female lengths: 7.9, 10.3, 9.4, 11.2 (n=4, y_bar =9.7, sY=1.4) 5 8
Estimate X-Y by constructing a 100(1-)% confidence interval.
Example 2. A manufacturing company has two different saws used for cutting columns.
On average, they suspect one saw (saw X) may be cutting columns shorter than the other
(saw Y). Does the following data provide significant evidence in support of this
X: 8.02 8.10 8.04 8.04 8.00 8.11 8.07 8.02 8.04 (m=9, x =8.05, sX=0.037)
Y: 8.04 8.04 8.10 8.06 8.08 8.10 8.07 8.08 8.06 (n=9, y =8.07, sX=0.022)
The following box plots confirm that the data supports the claim, but is it significant?
What is a point estimator for 1-2?
What is the sampling distribution?
Give the standardized estimator.
Case Confidence Interval Test Statistic
If both populations are assumed to have the same standard deviation, how does the
variance of the sampling distribution simplify?
What is a reasonable point estimator for the common sigma? What are the degrees of
Give an associated statistic that has a t distribution
If you do not assume the standard deviations of both populations are the same, how do
you estimate the variance of the sampling distribution? Does this lead to a test statistic
with a t distribution?
Case Confidence Interval Test Statistic When to use
Computations for example 1.
Computations for example 2. (Use the critical region approach.)
What is the P-value for the data in example 2?
Example 3. An experiment to compare the tension bond strength of polymer latex
modified mortar to unmodified mortar resulted in the following data
m 40, x 18.12, n 32, y 16.87
a. Assuming that 1 = 1.6 and 2 = 1.4, is this convincing evidence that the modified
mortar gives a higher average tension bond strength?
b. Compute the probability of type II error when the true average difference is 1?
c. Suppose the 40 specimens of modified mortar have already been collected. How
many unmodified specimens should be collected so that the power of the 0.05 level
test to detect a true average difference of 1 is 90%?
d. How would the conclusion of part a change if 1.6 and 1.4 were sample standard
Section 9.3 Paired Data.
Example 4. Makers of generic drugs must show that they do not differ significantly from
the reference drug that they imitate. One aspect they might differ in is their extent of
absorption in the blood. a. To study this, 10 subjects are randomly divided into two
groups. The first group is given only the reference drug. The second group is given only
the generic drug. The absorption of the given drug was measured.
Reference Drug Generic Drug
Estimate with 95% confidence the difference in mean absorption rates for the two drugs.
b. The same thing is being studied, but this time 10 subjects are randomly divided into
two groups. The first group was given the generic drug first, the second group was given
the reference drug first. In all cases, a washout period separated the two drugs so that the
first had disappeared from the blood before the subject took the second. The absorption
of each drug was measured.
Subject Reference Drug Generic Drug
1 1022 1284
2 1339 1930
3 2463 2120
4 2779 1613
5 2256 3052
6 1438 2549
7 1833 1310
8 3852 2254
9 1262 1964
10 4108 1755
Use the new data to estimate with 95% confidence the difference in mean absorption
rates for the two drugs.
Comments on Matched Pairs
In our example, the same subject was subjected to both treatments. When this cannot
be done, paired data can be obtained by matching subjects. For example, divide 10
subjects into 5 pairs according to blood pressure. In each pair, one person is given
reference drug and one generic drug.
In a paired design you have half the degrees of freedom as in a two sample design.
Pairing is still useful when enough variability in the response to the two treatments is
controlled by the pairing to overcome the loss of degrees of freedom.
Section 9.4. Comparing proportions.
Example 5. Mailing letters. Summarize data by giving sample proportion of all letters
that would be mailed. Construct a 95% confidence interval for the proportion of all Hope
students that would mail such a letter (consider the people in this class to be
representative of the population).
What is a point estimator for p1- p 2?
What is the sampling distribution?
Confidence Interval Test Statistic
Example 5b. Estimate the difference in the two population proportions of interest.
Example 5c. Is there significant evidence that p1> p 2?.
Suppose p1 = 0.75 and p2 = 0.35. Using equal sample sizes of 12, what is the power of
What sample size would be required so that the 0.05 level test detects the above specified
difference with a power of 90%?
Homework assignment for Chapter 9.
Section 9.1 1, 4, 5, 7, 10 (Hint on number 10: You first have to calculate s for each
sample The standard errors are 0.3 and 0.2 respectively, but the standard errors are s
divided by the square root of n).
Section 9.2 19, 20, 25, 27, 28
Section 9.3 37, 41, 45
Section 9.4 48, 49, 50, 51