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10.1 Inference for Mean Difference—Dependent Samples
10.2 Inference for Two Independent Means
10.3 Inference for Two Independent
Proportions
10.1 Inference for Mean Difference—Dependent Samples Objectives:
By the end of this section, I will be
able to…
1) Distinguish between independent samples and dependent samples.
2) Construct and interpret confidence intervals for the population mean difference for dependent samples.
3) Perform hypothesis tests for the population mean difference for dependent samples using the p-value method and the critical value method.
Independent Samples and Dependent Samples
Two samples are independent when the subjects selected for the first sample do not determine the subjects in the second sample.
Two samples are dependent when the subjects in the first sample determine the subjects in the second sample.
The data from dependent samples are called matched-pair or paired samples.
Example 10.1 - Dependent or independent sampling? Indicate whether each of the following experiments uses
an independent or dependent sampling method.
a. A study wished to compare the differences in price
between name-brand merchandise and store-brand
merchandise. Name-brand and store-brand items of the
same size were purchased from each of the following six
categories: paper towels, shampoo, cereal, ice cream,
peanut butter, and milk.
Example 10.1 continued
Solution
For a given store, each name-brand item in the first sample is associated with exactly one store-brand item of that size in the second sample.
Items in the first sample determine the items in the second sample
Example of dependent sampling
Example 10.1 - Dependent or independent sampling? Indicate whether each of the following experiments uses
an independent or dependent sampling method.
b. A study wished to compare traditional acupuncture
with usual clinical care for a certain type of lower-back
pain. The 241 subjects suffering from persistent
non-specific lower-back pain were randomly assigned to
receive either traditional acupuncture or the usual
clinical care. The results were measured at 12 and 24
months.
Example 10.1 continued
Solution
Randomly assigned to receive either of the two treatments
Thus, the subjects that received acupuncture did not determine those who received clinical care, and vice versa.
Example of independent sampling.
Example
Page 554, Problem 6
Solution
Example
Dependent Samples: Sample of the Differences
Set of matched-pair data obtained by taking dependent random samples of two populations
Find the difference to produce a random sample of the difference between the populations
Table 10.1 Statistics quiz scores of seven students before and after visiting the
Math Center
Difference: 16 13 14 18 14 11 12
Example 10.1 on page 545
Dependent Samples: Sample of the Differences
Sample mean of the sample differences:
147
12111418141316dx
FIGURE 10.1 Taking the differences reduces a two-sample problem to a single
sample of differences.
Point Estimates
Use known statistics to estimate unknown
parameters and report a single number
as the estimate
The value of the statistic is called the point estimate
Table 7.1 Point estimation: Use statistics to estimate unknown population parameters
Dependent Samples: Sample of the Differences
Sample mean of the sample differences:
Is a point estimate for the population mean of the sample differences:
dx
d
Dependent Samples: Sample of the Differences
Sample standard deviation of the sample differences:
Is a point estimate for the population standard deviation of the sample differences:
ds
d
Example
Page 554, Problem 10(a)
Solution
Example
First, calculate the differences:
Solution
Example
•First, enter the differences as a list of
data values
•Calculate mean and standard
deviation of the differences using:
1. STAT, CALC, 1:1-Var Stats
2. LIST, MATH, 3:mean( or
7:stdDev(
Directly from Calculator
Confidence Intervals: chapter 8
A confidence interval estimate of a parameter consists of an interval of numbers generated by a point estimate, together with an associated confidence level specifying the probability that the interval contains the parameter.
The meaning of a 100(1 confidence interval is as follows: If we take sample after sample for a very long time, then in the long run, the proportion of intervals that will contain the parameter μ will equal 100 100(1 - )%.
Dependent Samples: Sample of the Differences
Suppose we have a set of n paired differences:
With sample mean and sample standard deviation:
dd sx and
nddd ,...,, 21
Confidence interval for population mean of the differences
margin of error of the confidence interval
where is determined from the
t-distribution table using a given significance level and degrees of freedom
n
stE d
2/
2/t
1ndf
Confidence interval for population mean of the differences
lower bound of the confidence interval
upper bound of the confidence interval
Exd
Exd
Confidence interval for population mean of the differences
A 100(1- )% confidence interval for μd, the population mean of the differences, is given by
where:
),( ExExCI dd
n
stE d
2/
Confidence Interval Population
Mean Difference μd (Dependent Samples) continued
t interval applies whenever either of the following conditions is met:
Case 1: The population of differences is normal, or
Case 2: The sample size of differences is large
(n ≥ 30)
Example
Page 554, Problem 10(b)
Solution
and using significance level of 95% and t-distribution Table:
Example
571.22/t
Solution
Margin of error:
Example
4341.16
3663.1571.2
n
stE d
2/
Solution
Lower Bound:
Upper Bound:
Example
7674.04341.16667.0Exd
1008.24341.16667.0Exd
Confidence interval for population mean of the differences
A 95% confidence interval for μd, the population mean of the differences, is given by
Interpretation:
We are 95% confident that the true value of the population mean of the differences is between -0.7674 and 2.1008
)1008.2 ,7674.0(CI
T-Test for the Population Mean (chapter 9.4)
When population standard deviation is unknown, we may perform hypothesis testing for the mean with the t test using the p-value.
For the p-value method, we reject H0 if the p-value is less than .
Paired Sample t Test for the Population Mean: The p-Value Method
Set of matched-pair data
Dependent random samples of two populations
Find the difference to produce a random sample of the difference between the populations
Paired Sample t Test for the Population Mean: The p-Value Method Hypothesis Test Cases
Note: we always use 0 on the right side of the hypothesis test for the
population mean of the sample differences.
p-Values for t tests: use calculator for steps 2 and 3
First, find summary statistics: dd sx and
p-Values for t tests: calculator
dx
ds
0
Example
Page 555
Solution
Example
Solution
Example
Using calculator, the p-value is about 0.121
Solution
Example
Summary
Two samples are independent when the subjects selected for the first sample do not determine the subjects in the second sample.
Two samples are dependent when the subjects in the first sample determine the subjects in the second sample.
The data from dependent samples are called matched-pair or paired samples.
Summary
The key concept in this section is that we consider the differences of matched pair data as a sample, and perform inference on this sample of differences.
A 100(1- )% confidence interval for μd, the population mean of the differences, is given by where xd and sd represent
the sample mean and sample standard deviation of the differences, respectively, of the set of n paired differences, d1, d2, d3, . . . , dn, and where tα/2 is based on n - 1 degrees of freedom.
/2 /d dx t s n
Summary
The paired sample t test for the population mean of the differences μd can be used under either of the following conditions:
Case 1: the population is normal, or
Case 2: the sample size is large (n ≥ 30).
The test may be carried out using the p-value method.
10.2 Inference for Two Independent Means Objectives:
By the end of this section, I will be
able to…
1)Describe the sampling distribution of
2)Compute and interpret t intervals for μ1-μ2
3)Perform and interpret t tests about μ1-μ2
4)Use confidence intervals for μ1-μ2 to perform two-tailed hypothesis tests about μ1-μ2
1 2x x
Sampling Distribution of x1-x2
Random samples drawn independently from populations with population means μ1 and μ2
and either
Case 1:
The two populations are normally distributed, or
Case 2:
The two sample sizes are large (at least 30), then the quantity
Example Page 569
Solution
Example
x1, s1, and n1 represent the mean, standard
deviation, and sample size of the sample
taken from population 1
x2, s2, and n2 represent the mean, standard
deviation, and sample size of the sample
taken from population 2
Sampling Distribution of x1-x2
Standard Error of x1-x2
Standard error of the statistic is
It measures the size of the typical error in using to measure μ1- μ2.
21
2 2
1 2
1 2
x x
s ss
n n
1 2x xs 1 2x x
1 2x x
Sampling Distribution of x1-x2
continued
Approximately a t distribution Degrees of freedom equal to the smaller of
n1 - 1 and n2 – 1
21
1 2 1 2 1 2 1 2
2 2
1 2
1 2
x x
x x x xt
s s s
n n
margin of error of the confidence interval
where is determined from the
t-distribution table using confidence level
and degrees of freedom df equal to the smaller of
2
2
2
1
2
12/
n
s
n
stE
2/t
Confidence Interval for μ1-μ2
1 and 1 21 nn
lower bound of the confidence interval
upper bound of the confidence interval
Exx 21
Exx 21
Confidence Interval for μ1-μ2
A 100(1- )% confidence interval for is given by
where:
Confidence Interval for μ1-μ2
21
),( 2121 ExxExxCI
2
2
2
1
2
12/
n
s
n
stE
Confidence Interval for μ1-μ2
continued
The t interval applies whenever either of the following conditions is met:
Case 1:
Both populations are normally distributed
Case 2:
Both sample sizes are large.
Example Page 570
Solution
Example
Solution
Example
Solution
Example
Solution
Lower Bound:
Upper Bound:
Example
3488.11621 Exx
3488.221 Exx
Confidence interval for population mean of the differences
A 90% confidence interval for μd, the population mean of the differences, is given by
Interpretation:
We are 90% confident that the true value of the difference in the population mean math scores of students from US and Hong Kong is between -116.35 and 2.35
)3488.2 ,3488.116(CI
Hypothesis Test for the Difference in Two Population Means
p-Value Method
Step 1
State the hypotheses and the rejection rule.
Use one of the forms from Table 10.14 page 561.
Table 10.14 Three possible forms for the hypotheses for a test about µ – µ 1 2
Hypothesis Test for the Difference in Two Population Means
p-Value Method
Step 1 (cont.)
Clearly state the meaning of μ1 and μ2.
The rejection rule is Reject H0 if the p-value is less than .
Hypothesis Test for the Difference in Two Population Means continued
Step 2
Find tdata.
which follows an approximate t distribution with degrees of freedom the smaller of n1-1 and n2-1.
21
1 2 1 2
2 2
1 2
1 2
data
x x
x x x xt
s s s
n n
Hypothesis Test for the Difference in Two Population Means continued
Step 3
Find the p-value.
p-Values for t tests: use calculator for steps 2 and 3
p-Values for t tests: use calculator for steps 2 and 3
Hypothesis Test for the Difference in Two Population Means continued
Step 4
State the conclusion and interpretation.
Compare the p-value with .
Example Page 571
Solution
Example
students coachedfor t improvemen
score SATmean population1
students coached-nonfor t improvemen
score SATmean population2
Solution
Use calculator, 2-SampTTest with sample size n=100 and:
p-value=0.1552
Example
100 59, 29, 111 nsx
100 52, 21, 111 nsx
Solution
Example
Summary
Section 10.2 examines inferential methods for μ1-μ2, the difference between the means of two independent populations.
The section begins with a discussion of the sampling distribution of x1- x2, which underlies the inference in the remainder of the section.
100(1- )% t confidence intervals for μ1-μ2 are developed and illustrated.
Summary
Two-sample t tests are discussed.
These hypothesis tests may be carried out using the p-value.
10.3 Inference for Two Independent Proportions Objectives:
By the end of this section, I will be
able to…
1) Understand the sampling distribution of
p1 - p2.
2) Compute and interpret confidence intervals for p1 - p2.
3) Perform and interpret hypothesis tests for p1 - p2.
ˆ ˆ
Sampling Distribution of p1 - p2
Independent random samples from two populations
There are successful outcomes in sample one and sample size
There are successful outcomes in sample one and sample size
Sample proportions are:
ˆ and ˆ2
22
1
11
n
xp
n
xp
1x
1n
2x
2n
Sampling Distribution of p1 - p2
The population proportions from each sample are
21 and pp
Sampling Distribution of p1 - p2
Let
1 2
1 2 1 2
ˆ ˆ
1 2 1 2
1 1 2 2
1 2
ˆ ˆ
ˆ ˆ
p p
p p p pZ
p p p p
p q p q
n n
1 and 1 2211 pqpq
Sampling Distribution of p1 - p2
continued
Has an approximately standard normal distribution when the following conditions are satisfied:
x1 ≥ 5, (n1 - x1) ≥ 5, x2 ≥ 5, (n2 - x2) ≥ 5
ˆ and ˆ :NOTE 222111 pnxpnx
)ˆ1( )( and )ˆ1( )( 22221111 pnxnpnxn
Standard Error of p1 - p2
Standard error of the statistic p1 - p2
Where q1 = 1 - p1 and q2 = 1 - p2.
The standard error measures the size
of the typical error in using p1 - p2 to
estimate p1 - p2.
1 2ˆ ˆp ps
1 1 2 2
1 2ˆ ˆ
1 2
ˆ ˆ ˆ ˆp p
p q p qs
n n
1 2ˆ ˆp ps
Confidence Interval for p1 – p2
Margin of Error E
100(1- )% confidence interval for p1 - p2 is
1 2ˆ ˆ/2 /2
1 1 2 2/2
1 2
standard error
ˆ ˆ ˆ ˆ
p pE Z Z s
p q p qZ
n n
A 100(1- )% confidence interval for is given by
where:
21 pp
)ˆˆ,ˆˆ( 2121 EppEppCI
Confidence Interval for p1 - p2
2
22
1
112/
ˆˆˆˆ
n
qp
n
qpZE
Confidence Interval for p1 - p2
Independent random samples taken from
two populations with population proportions
p1 and p2,
Confidence Interval for p1 - p2
continued
Where p1 and n1 represent the sample proportion and sample size of the sample taken from population 1 with population proportion p1
p2 and n2 represent the sample proportion and sample size of the sample taken from population 2 with population proportion p2; the samples are drawn independently
Required conditions:
x1 ≥ 5, (n1 - x1) ≥ 5, x2 ≥ 5,
and (n2 - x2) ≥ 5.
Example Page 583
Example Solution
Alaska
Arizona
Example Solution
Example Solution
Example
The point estimate of the difference in sample proportions lies
within 0.0030 of the true value of the difference in population
proportions with 99% confidence.
Solution
Example Solution
)0285.0,0345.0(
)ˆˆ,ˆˆ( 2121 EppEppCI
Hypothesis Test for the Difference in Two Population Proportions: p-Value Method
Two independent random samples
Taken from two populations
Population proportions p1 and p2
Required conditions:
x1 ≥ 5, (n1 - x1) ≥ 5, x2 ≥ 5,
and (n2 - x2) ≥ 5.
Table 10.19 Three possible forms for the hypotheses for a test about p – p 1 2
p-Values for t tests: use calculator for steps 2 and 3
FIGURE 10.20
Example Page 583
Solution
Example
Ohio
New Jersey
Solution
Example
Solution
Example
Solution
Example
Summary
The section discusses inferential methods for p1 - p2, the difference between the proportions of two independent populations.
The section begins with a discussion of the sampling distribution of p1 - p2, which underlies the inference in the remainder of the section.
Summary
100(1- )% Z confidence intervals for p1 - p2 are developed and illustrated.
Two-sample Z tests are discussed.
These hypothesis tests may be carried out using either the p-value method or the critical value method.