Overview

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.