Chapter 10 Inferences from Two Samples

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Chapter 10 Inferences from Two Samples

• Inferences about Two Means: Independent and Large Samples

• Inferences about Two Means: Independent and Small Samples

• Inferences about Two Means: Matched Pairs

• Inferences about Two Proportions

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Overview

There are many important and meaningful situations in which it becomes necessary

to compare two sets of sample data.

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10-2

Inferences about Two Means:Independent and

Large Samples

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Definitions

Two Samples: IndependentThe sample values selected from one population are not related or somehow paired with the sample values selected from the other population.

If the values in one sample are related to the values in the other sample, the samples are dependent. Such samples are often referred to as matched pairs or paired samples.

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Assumptions

1. The two samples are independent.

2. The two sample sizes are large. That is, n1 > 30 and n2 > 30.

3. Both samples are simple random samples.

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Characteristics of theSampling Distribution XA-XB

• The mean of the sampling distribution of all possible XA-XB is µA- µB.

• The standard deviation of the sampling distribution of all the possible values of

BA XX

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Hypothesis Tests

Test Statistic for Two Means: Independent and Large Samples

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Hypothesis Tests


(x1 - x2) - (µ1 - µ2)z =n1 n2

+1

. 2

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Hypothesis Tests


and If and are not known, use s1 and s2 in their places. provided that both samples are large.

P-value: Use the computed value of the test

statistic z, and find the P-value .

Critical values: Based on the significance level , find critical values .

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Coke Versus PepsiSample statistics are shown. Use the 0.01 significance level to test the claim that the mean weight of regular Coke is different from the mean weight of regular Pepsi.

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Coke Versus PepsiSample statistics are shown. Use the 0.01 significance level to test the claim that the mean weight of regular Coke is different from the mean weight of regular Pepsi.

Regular Coke Regular Pepsi

n 36 36

x 0.81682 0.82410

s 0.007507 0.005701

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Coke Versus Pepsi

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Claim: 1 2

Ho : 1 = 2

H1 : 1 2

= 0.01

Coke Versus Pepsi

Fail to reject H0Reject H0Reject H0

Z = - 2.575 Z = 2.5751 - = 0

or Z = 0

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2

(x1 - x2) - (µ1 - µ2)z =n1 n2

+1

. 2

2

Coke Versus Pepsi

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(0.81682 - 0.82410) - 0z =36

+

Coke Versus Pepsi

0.0075707 2 0.005701 2

36

= - 4.63

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Claim: 1 2

Ho : 1 = 2

H1 : 1 2

= 0.01

Coke Versus Pepsi

Fail to reject H0Reject H0 Reject H0

Z = - 2.575 Z = 2.5751 - = 0

or Z = 0

sample data:z = - 4.63

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Claim: 1 2

Ho : 1 = 2

H1 : 1 2

= 0.01

Coke Versus Pepsi

Fail to reject H0Reject H0 Reject H0

Z = - 2.575 Z = 2.5751 - = 0

or Z = 0

sample data:z = - 4.63

Reject Null

There is significant evidence to support the claim that there is a difference

between the mean weight of Coke and the mean weight of Pepsi.

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Confidence Intervals

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(x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E

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(x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E

n1 n2

+1 2where E = z

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(x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E

where E = z n1 n2

+1 2

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For Coke versus Pepsi, x1 - x2 = .00728, and z

Find an 80% confidence interval for the difference for Cokeand Pepsi.

.00728 + 1.28(.00157) = (.00527, .00929)

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t-Distribution Model

•The degrees of freedom are nA+ nB – 2•A pooled variance is the weighted mean of the sample variances. and is used if the the data is not normally distributed.

2nn

s1)(ns1)(ns

BA

2BB

2AA2

p

2

(x1 - x2) - (µ1 - µ2)t =n1 n2

+1

. 2

2

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Two groups were tested to see whether calcium reduces bloodpressure. The following data was collected. Is there evidence atthe .1 level that calcium reduces blood pressure?

Group 1 (calcium) 7 -4 18 17 -3 -5 1 10 11 –2Group 2 (placebo) -1 12 -1 -3 3 -5 5 2 -11 -1 -3

Group Treatment n x s 1 Calcium 10 5.00 8.743 2 Placebo 11 -.273 5.901

1. HO: µ1 - µ2 > 0 HA: µ1 - µ2 < 0

2. One tail t test, n < 30 11 + 10 – 2 = 19 d.f.

3. tcritical = -1.328

1.604

115.901

108.743

.273)(5.00

ns

ns

xxt 4.

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2

22

1

21

21

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5. There is not enough evidence at the .1 level that calcium reduces blood pressure.

-1.328 1.604

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Inferences about Two Proportions

Assumptions

1. We have proportions from two independent simple random samples.

2. For both samples, the conditions np 5 and nq 5 are satisfied.

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Notation for Two ProportionsFor population 1, we let:

p1 = population proportion

n1 = size of the sample

x1 = number of successes in the sample

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p1 = x1/n1 (the sample proportion)^

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p1 = x1/n1 (the sample proportion)

q1 = 1 - p

1

^^

^

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π1 = population proportion



p1 = x1/n1 (the sample proportion)

q1 = 1 - p

1

to π2, n

2 , x

2 , p

2. and q

2 , which come from

population 2.

The corresponding meanings are attached

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Test Statistic for Two Proportions

For H0: π1 = π

2 , H0: π1

π2 , H0: π1

π2

HA:π

1 π

2 , HA: π

1 < π

2 , HA: π

1> π

2

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where π1 - π

2 = 0 (assumed in the null hypothesis)


For H0: p1 = p

2 , H0: p1

p2 , H0: p1

p2

H1: p1

p2 , H1: p1

< p2 , H1: p

1> p

2

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x1

n1

x2

n2

p1

p2

where p1 - p

2 = 0 (assumed in the null hypothesis)

= =and

For H0: p1 = p

2 , H0: p1

p2 , H0: p1

p2

H1: p1

p2 , H1: p1

< p2 , H1: p

1> p

2

1n

)P(1P

1n

)P(1PSE

B

BB

A

AA

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(p1 - p

2 ) - E < (π

1 - π

2 ) < (p

1 - p

2 ) + E

Confidence Interval Estimate of π1 - π

2

If 0 is not in the interval, one may beC% confident that the two population proportions are different.

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A sample of households in urban and rural homes displayed thefollowing data for preference of artificial or natural Christmas trees:

Population n X p = X/n 1(urban) 261 89 .341 2(rural 160 64 .400

Is there a difference in preference between urban and rural homes?with a confidence interval of 90%?

1n

)P(1P

1n

)P(1PSE

B

BB

A

AA

.048159

.400(.600)

260

.341(.659)SE

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SEZ)( critical21

.04851.645.400)(.341

(-.139, .021)

We are 90% confident that the difference in proportions is between-.14 and .02. Because the interval contains 0, we are not confidentthat either group has a stronger preference for natural trees thanthe other group.

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Assumptions

1. The sample data consist of matched pairs.

2. The samples are simple random samples.

3. If the number of pairs of sample data is small (n 30), then the population of differences in the paired values must be approximately normally distributed.

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Notation for Matched Pairs

µd = mean value of the differences d for the

population of paired data

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sd = standard deviation of the differences d for

the paired sample data

n = number of pairs of data.

µd = mean value of the differences d for the

population of paired data

d = mean value of the differences d for the

paired sample data (equal to the mean of the x - y values)

Notation for Matched Pairs

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Test Statistic for Matched Pairs of Sample Data

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t = d - µdsd

n

Test Statistic for Matched Pairs of Sample Data

where degrees of freedom = n - 1

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Critical Values

If n 30, critical values are found in Table A-4 (t-distribution).

If n > 30, critical values are found in Table A- 2 (normal distribution).

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d - ME < µd < d + ME

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degrees of freedom = n -1

d - ME < µd < d + ME

where ME = t sd

n

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How Much Do Male Statistics Students Exaggerate Their Heights?

Using the sample data from Table 8-1 with the outlier excluded, construct a 95% confidence interval estimate of d, which is the mean of the differences between reported heights and measured heights of male statistics students.

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Reported and Measured Heights (in inches) of Male Statistics Students

Student A B C D E F G H I J K L

Reported 68 74 82.25 66.5 69 68 71 70 70 67 68 70 Height

Measured 66.8 73.9 74.3 66.1 67.2 67.9 69.4 69.9 68.6 67.9 67.6 68.8Height

Difference 1.2 0.1 7.95 0.4 1.8 0.1 1.6 0.1 1.4 -0.9 0.4 1.2

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d = 1.279

s = 2.243

n = 12

t = 2.201 (found from Table A-3 with 11 degrees of freedom and 0.05 in two tails)


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E = t sd

n


E = (2.201)( )2.244

12

= 1.426

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1.279 - 1.464 < µd < 1.279 + 1.464

In the long run, 95% o f such samples will lead to confidence intervals that actually do contain the true population mean of the differences. Since

the interval does contain 0, the true value of µd is

not significantly different from 0. There is not sufficient evidence to support the claim that there is a difference between the reported heights and the measured heights of male statistics students.

Chapter 10 Inferences from Two Samples

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