Chapter 10 Statistical Inference for Two Samples

More than one but less than three!

Chapter 10B< X <

Things Statistical

• A special case of the two-sample t-tests occurs when the observations on the two populations of interest are collected in pairs. They are not independent observations.

• Each pair of observations, say (X1j , X2j ), is taken under homogeneous conditions.

•Each observation in one sample is paired naturally or by design with an observation in the other sample.

•The test procedure consists of analyzing the differences between each the pairs.

10-4 Paired t -Test

10-4 Paired t-Test

1 2

1 2 ; 1,...,i i i

D X X

d x x i n

Example 10-9

Example 10-9

Example 10-9

Paired Versus Unpaired Comparisons

A Confidence Interval for D

Definition

Example 10-10

Example 10-10

1.21

12.68 / 14d

d

s

Example 10-10

10-5.1 The F Distribution

10-5 Inferences on the Variances of Two Normal Populations

We wish to test the hypotheses:

The development of a test procedure for these hypotheses requires the F distribution.

10-5.1 The F Distribution

More F Distribution

Look at this…

The lower-tail percentage points f-1,u, can be found as follows.

10-5.2 Hypothesis Tests on the Ratio of Two Variances

10-5.2 Hypothesis Tests on the Ratio of Two Variances

2 21 12 22 2

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SF

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Example 10-11

Example 10-11

Example 10-11

10-5.3 Type II Error and Choice of Sample Size

Example 10-12

10-5.4 Confidence Interval on the Ratio of Two Variances

1,1,2/22

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Example 10-13

Example 10-13

10-6 Large-Sample Test on the Difference in Population Proportions

0 1 2

1 20

1 2

Null hypothesis: :

ˆ ˆTest Statistic:

1 1ˆ ˆ(1 )

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Alternate Hypothesis

Rejection Criterion

1 2

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Example 10-14

A depressed stat studentwho was unable to reject his null hypothesis. Very disturbed.

Example 10-14

Example 10-14

Minitab Output for Example 10-14

Inferences on Two Population Proportions- beta errors

Unless p1=p2, you cannot use the estimate of the standard deviation as in the hypothesis tests. Instead we use,

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10-6.2 Type II Error and Choice of Sample Size

10-6.2 Type II Error and Choice of Sample Size

Confidence Interval on the Difference in the Population Proportions

Example 10-15

8-7

Example 10-15

Summary Table and Road Map for Inference Procedures for Two Samples

Summary Table and Road Map for Inference Procedures for Two Samples

We have concluded our discussion on two samples

Should we not go to three or more sample tests?

A statistics student countingthe number of samples…