Chapter 11Inferences on Two Samples
Ch 11.1 Inference about Two Population Proportions
Objective A :Distinguish between Independent and Dependent Sampling
Example 1:
Determine whether each sampling method is independent or dependent.
(a) Test scores of the same students in English and Math.
Since the same students performed both tests, this sampling method
is dependent.
(b) The effectiveness of two different diets on two different groups of
individuals.
Since two different groups of individuals performed two different
tests, this sampling method is independent.
Objective B :Test Hypotheses or Confidence Intervals Regarding Two Proportions from
Independent Samples
Example 1:
The drug Prevnar is a vaccine meant to prevent certain types of bacterial
meningitis. It is typically administered to infants starting around 2 months
of age. In randomized, double-blind clinical trials of Prevnar, infants were
randomly divided into two groups. Subjects in group 1 received Prevnar,
while subjects in group 2 received a control vaccine. After the second dose,
137 of 452 subjects in the experimental group (group 1) experienced drowsiness
as a side effect. After the second dose, 31 of 99 subjects in the control group
(group 2) experienced drowsiness as a side effect.Does the evidence suggest
that a lower proportion of subjects in group 1 experienced drowsiness as a side
effect than subjects in group 2 at the   0.05 level of significance?
(a) Setup
Independent Samples for Two Proportions
Group 1
Group 2
x 137
x
31
pˆ 1  1 
 0.3031
pˆ 2  2 
 0.3131
n1 452
n2 99
Ho: p1  p2 ---> p1  p2  0
H1:
p1  p2 ---> p1  p2  0
(b) P  value
We must verify the requirements to perform the hypothesis test between
two population proportions.
Stat --> Proportion Stats --> Two Sample --> With Summary -->
Input the following, then hit Compute!
StatCrunch Results:
P-value = 0.4221
(c) Conclusion
Since the P-value (0.4221) is not less than 0.05, do not reject Ho.
There is not enough evidence to support that a lower proportion
of subjects in group 1 experienced drowsiness as a side effect than
subjects in group 2.
Example 2:
The body mass index (BMI) of an individual is one measure that is used to
judge whether an individual is overweight or not. A BMI between 20 and
25 indicates that one is at a normal weight. In a survey of 750 men and 750
women, the Gallup organization found that 203 men and 270 women were
normal weight. Construct a 90% confidence interval to gauge whether there
is a difference in the proportion of men and women who are normal weight.
Interpret the interval.
C. I. for p1  p2
We must verify the requirements for constructing a confidence interval for the
difference between two population proportions.
Stat --> Proportion Stats --> Two Sample --> With Summary -->
Input the following, then hit Compute!
StatCrunch Results:
90% confidence interval for pm  p w is (-0.0933, -0.0134).
We are 90% confidence that the difference in the proportion of men and
women that are normal weight is between - 9.3% and -1.3%. Because the
confidence interval limits do not contain 0, there is a significant difference
between the two proportions. Because the confidence interval includes only
negative values, it appears that the normal weight rate for men is less than
the rate for women (i.e. a higher proportion of males are not normal weight).
Ch 11.2Inference about Two Means: Dependent Samples
Objective A :Test Hypotheses or Confidence Intervals about the Population Mean
Difference of Matched-Pairs Data
Example 1:
In an experiment conducted online at the University of Mississippi, study
participants are asked to react to a stimulus. In one experiment, the
participant must press a key on seeing a blue screen. Reaction time (in
seconds_ to press the key is measured. The same person is then asked
to press a key on seeing a red screen, again with reaction time measured.
The results for six randomly sampled study participants are as follows:
(a) Why are these matched-pairs data?
The same participants performed the reaction time for seeing a blue
screen and a red screen.
(b) Is the reaction time to the blue stimulus different from the reaction
time to the red stimulus at the   0.01 level of significance?
Note: A normal probability plot and boxplot of the data indicate that the
differences are approximately normally distributed with no outliers.
Dependent Samples (Matched-Pairs Design)
--> Difference in reaction time
= reaction time seeing a blue screen - reaction time seeing a red screen
Setup: Ho:  d  0
H1:
d  0
Open StatCrunch
--> Input reaction time for seeing a blue screen in column 1 (Var1) and
reaction time for seeing a red screen in column 2 (Var 2) --> T Stats -->
Paired --> Input the following -->Compute!
StatCrunch Results:
Since the p-value (0.2466) is not less than  (0.01), do not reject Ho.
There is not sufficient evidence to support the claim that reaction time
to the blue stimulus is different from the reaction time to the red stimulus.
(c) Construct a 99% confidence interval about the population mean difference.
Interpret your results.
Stat --> T Stats --> Paired --> Input the following --> Compute!
StatCrunch Results:
99% confidence interval for  d is (-0.193, 0.379).
We are 99% confidence that the difference in the reaction time between the
blue stimulus and red stimulus is between -0.193 second and 0.379 second.
The confidence interval includes 0, which suggests that the mean of the
differences could be 0 and there is no change in reaction time for blue
stimulus versus red stimulus.
Ch 11.3Inference about Two Means: Independent Samples
Objective A :Test Hypotheses or Confidence Intervals regarding the Difference of Two
Independent Means
If we can assume  1   2 , use t  distribution with POOLED standard error.
In general, we use t  distribution without POOLED standard error unless instructed
otherwise.
Example 1:
(a) The normal probability plots indicated the samples came from the populations that
are normally distributed. The boxplots indicated the samples had no outliers. Assuming
the samples were randomly selected and each sample size is no more than 5% of the
population size, Welch's t-test can be used.
(b) Independent Samples for Two Means
Ho:  f   m --->  f   m  0
H1:  f   m ---->  f   m  0
The example was obtained from the Chapter 11 Section 3 HW Exercise 14.
We retreive the built-in Ch11_3_14 data and compute the results through StatCrunch.
StatCrunch --> data sets from your textbook --> chapter 11 --> 11_3_14_txt -->
Stat --> T Stats --> Two Sample --> With Data --> Input the following and uncheck Pool
variances under Calculation options: --> Compute!
StatCrunch Results:
Since p-value is not less than
 (0.05), do not reject Ho. There is not enough evidence
to support the claim that there is a difference in the reaction times of males and
females.
(c) Graph --> Boxplot --> Select Female Students, press the control key then select Male Students
--> Input the following --> Compute!
The above boxplots support the finding from part (b).
Example 2:
Independent Samples for Two Means
Construct and interpret a 90% confidence intervals for  G   A .
Stat --> T Stats --> Two Sample --> With Summary --> Input the following and uncheck Pool
variances under Calculation options: --> Compute!
StatCrunch Results:
We are 90% confident that the mean difference in cooling time between filled clear glass bottles and
filled aluminum bottles is between 38.13 minutes and 44.67 minutes. Since the confidence interval
does not include 0, the two population means do not appear to be equal. The confidence interval
includes only positive values, which suggests that the mean cooling time for chilling a bottle of beer in a
clear glass bottle is greater than in an aluminum bottle.
Ch 11.4Inference about Two Population Standard Deviations
Objective A : Fisher’s F  distribution
Objective B : Test Hypotheses regarding Two Population Standard Deviations
Example 1:
Assume that the populations are normally distributed.
Concepts for Hypothesis Tests with Two Variances
s12
F  2 where s12 is the larger of the two sample variances.
s2
If the two populations really do have equal variances, then the ratio of
s12
should be close to 1 because
s 22
s12 and s 22 tend to be close in value. If the two populations do not have equal variances, then the ratio of
s12
will be bigger than 1 by selecting s12 be the larger sample variance. Consequently, a large value of F
2
s2
will be evidence against  12   22 .
First, we need to identify what is s12 . s12 is the larger of the two sample variance.
In this problem, s12  9.2 2  84.64 and s 22  8.6 2  73.96
Stat --> Variance Stats --> Two Sample Homogeneity --> With Summary ---> Input the following
---> Compute!
StatCrunch Results:
Since the P-value (0.7659) is not less than
  0.1, do not reject Ho:  12   22 .
There is not sufficient evidence to conclude that  12   22 <-->  1   2
Example 2:
Identify s12 --> s12 is the larger of the two sample variances.
s12  532  2809 ---> n1  260
s 22  34 2  1156 ---> n2  269
Stat --> Variance Stats --> Two Sample Homogeneity --> With Summary ---> Input the following
---> Compute!
StatCrunch Results:
Since the P-value (<0.0001) is less than
  0.05 , reject Ho:  12   22 .
There is sufficient evidence to conclude that the standard deviation walking speed is different between
the two groups.