Section 9.2-1 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Lecture Slides Elementary Statistics Twelfth Edition and the
Triola Statistics Series by Mario F. Triola
Slide 2
Section 9.2-2 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Chapter 9 Inferences from Two Samples 9-1 Review and Preview
9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two
Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard
Deviations
Slide 3
Section 9.2-3 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Key Concept In this section we present methods for (1) testing
a claim made about two population proportions and (2) constructing
a confidence interval estimate of the difference between the two
population proportions. This section is based on proportions, but
we can use the same methods for dealing with probabilities or the
decimal equivalents of percentages.
Slide 4
Section 9.2-4 Copyright 2014, 2012, 2010 Pearson Education,
Inc. (the sample proportion) The corresponding notations apply to
which come from population 2. Notation for Two Proportions For
population 1, we let: = population proportion = size of the sample
= number of successes in the sample
Slide 5
Section 9.2-5 Copyright 2014, 2012, 2010 Pearson Education,
Inc. The pooled sample proportion is given by: Pooled Sample
Proportion
Slide 6
Section 9.2-6 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Requirements 1. We have proportions from two independent
simple random samples. 2.For each of the two samples, the number of
successes is at least 5 and the number of failures is at least 5. n
5 n(1- ) 5 For each of the two samples
Slide 7
Section 9.2-7 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Test Statistic for Two Proportions For where (assumed in the
null hypothesis)
Slide 8
Section 9.2-8 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Testing Two Proportions P-value: P-values are provided by, use
Table A-2. Critical values: Use Table A-2. (Based on the
significance level , find critical values by using the procedures
introduced in Section 8-2 in the text.)
Slide 9
Section 9.2-9 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Confidence Interval Estimate of p 1 p 2
Slide 10
Section 9.2-10 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example Do people having different spending habits depending
on the type of money they have? 89 undergraduates were randomly
assigned to two groups and were given a choice of keeping the money
or buying gum or mints. The claim is that money in large
denominations is less likely to be spent relative to an equivalent
amount in many smaller denominations. Lets test the claim at the
0.05 significance level.
Slide 11
Section 9.2-11 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example Below are the sample data and summary statistics:
Group 1Group 2 Subjects Given $1 Bill Subjects Given 4 Quarters
Spent the moneyx 1 = 12x 2 = 27 Subjects in groupn 1 = 46n 2 =
43
Slide 12
Section 9.2-12 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example Requirement Check: 1.The 89 subjects were randomly
assigned to two groups, so we consider these independent random
samples. 2.The subjects given the $1 bill include 12 who spent it
and 34 who did not. The subjects given the quarters include 27 who
spent it and 16 who did not. All counts are above 5, so the
requirements are all met.
Slide 13
Section 9.2-13 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example Step 1: The claim that money in large denominations is
less likely to be spent can be expressed as p 1 < p 2. Step 2:
If p 1 < p 2 is false, then p 1 p 2. Step 3: The hypotheses can
be written as:
Slide 14
Section 9.2-14 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example Step 4: The significance level is = 0.05. Step 5: We
will use the normal distribution to run the test with:
Slide 15
Section 9.2-15 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example Step 6: Calculate the value of the test
statistic:
Slide 16
Section 9.2-16 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example Step 6: This is a left-tailed test, so the P-value is
the area to the left of the test statistic z = 3.49, or 0.0002. The
critical value is also shown below.
Slide 17
Section 9.2-17 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example Step 7: Because the P-value of 0.0002 is less than the
significance level of = 0.05, reject the null hypothesis. There is
sufficient evidence to support the claim that people with money in
large denominations are less likely to spend relative to people
with money in smaller denominations. It should be noted that the
subjects were all undergraduates and care should be taken before
generalizing the results to the general population.
Slide 18
Section 9.2-18 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example We can also construct a confidence interval to
estimate the difference between the population proportions.
Caution: The confidence interval uses standard deviations based on
estimated values of the population proportions, and consequently, a
conclusion based on a confidence interval might be different from a
conclusion based on a hypothesis test.
Slide 19
Section 9.2-19 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example Construct a 90% confidence interval estimate of the
difference between the two population proportions. What does the
result suggest about our claim about people spending large
denominations relative to spending small denominations?
Slide 20
Section 9.2-20 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example
Slide 21
Section 9.2-21 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Example The confidence interval limits do not include 0,
implying that there is a significant difference between the two
proportions. There does appear to be sufficient evidence to support
the claim that money in large denominations is less likely to be
spent relative to an equivalent amount in many smaller
denominations.
Slide 22
Section 9.2-22 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Why Do the Procedures of This Section Work? The distribution
of can be approximated by a normal distribution with mean p 1, and
standard deviation: The difference can be approximated by a normal
distribution with mean p 1 p 2 and variance The variance of the
differences between two independent random variables is the sum of
their individual variances.
Slide 23
Section 9.2-23 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Why Do the Procedures of This Section Work? The preceding
variance leads to We now know that the distribution of is
approximately normal, with mean and standard deviation as shown
above, so the z test statistic has the form given earlier.
Slide 24
Section 9.2-24 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Why Do the Procedures of This Section Work? When constructing
the confidence interval estimate of the difference between two
proportions, we dont assume that the two proportions are equal, and
we estimate the standard deviation as
Slide 25
Section 9.2-25 Copyright 2014, 2012, 2010 Pearson Education,
Inc. Why Do the Procedures of This Section Work? In the test
statistic use the positive and negative values of z (for two tails)
and solve for The results are the limits of the confidence interval
given earlier.
Slide 26
Section 9.2-26 Copyright 2014, 2012, 2010 Pearson Education,
Inc. #8 The herb ginkgo biloba is commonly used as a treatment to
prevent dementia. In a study of the effectiveness of this
treatment, 1545 elderly subjects were ginkgo and 1524 elderly
subjects were given a placebo. Among those in the ginkgo treatment
group, 246 later developed dementia, and among those in the placebo
group, 277 later develop dementia. We want to use a 0.01
significance level to test the claim that ginkgo is effective in
preventing dementia. a)Test the claim using a hypothesis test.
b)Test the claim by constructing an appropriate confidence
interval