Chapter 3 Tests of Hypotheses for Two Samples

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Chapter 3 Tests of Hypotheses for Two Samples
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Agenda

• Inference for a difference in means (variance
  known)
• Inference for the difference in means ( variance
  unknown)
• Paired t-test
• Inference on the variances
• Inference on the proportions

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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
Two independent populations.
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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
Assumptions
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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
1. Hypothesis Tests
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Example 1
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Example 1
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Example 1
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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
2. Choice of Sample Size
Use of Operating Characteristic Curves
Two-sided alternative
One-sided alternative
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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
2. Choice of Sample Size
Sample Size Formulas
Two-sided alternative
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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
2. Choice of Sample Size
Sample Size Formulas
One-sided alternative
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Example 2
1
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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
3. Identifying Cause and Effect

• When statistical significance is observed in a
  randomized experiment, the experimenter can be
  confident in the conclusion that it was the
  difference in treatments that resulted in the
  difference in response.
• That is, we can be confident that a
  cause-and-effect relationship has been found.

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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
4. Confidence Interval
Definition
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Example 3
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Example 3
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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
Choice of Sample Size
Where E limit error in estimating µ1 µ2 by
the different of sample means at 100(1- ?)
confidence
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Inference for a Difference in Means of Two
Normal Distributions, Variances Known
One-Sided Confidence Bounds
Upper Confidence Bound
Lower Confidence Bound
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Inference for a Difference in Means of Two
Normal Distributions, Variances Unknown
1. Hypotheses Tests
Case 1
We wish to test
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Inference for a Difference in Means of Two
Normal Distributions, Variances Unknown
1. Hypotheses Tests
Case 1
The pooled estimator of ?2
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Inference for a Difference in Means of Two
Normal Distributions, Variances Unknown
1. Hypotheses Tests
Case 1
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Inference for a Difference in Means of Two
Normal Distributions, Variances Unknown
Definition The Two-Sample or Pooled t-Test
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Example 4
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Example 4
4
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Example 4
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Example 4
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Normal probability plot and comparative box plot
for the catalyst yield data in Example 4. (a)
Normal probability plot, (b) Box plots.

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Minitab Practice for Example 4

• Data file example3_4.xls
• Menu ? Stat ? Basic statistics ? 2 sample t ?
  select Samples in different columns ?check
  Assume equal variances ?options
• Confident level 95
• Test mean 0.0
• Alternative not equal

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Inference for a Difference in Means of Two
Normal Distributions, Variances Unknown
1. Hypotheses Tests for a Difference in Means,
Variances Unknown
Case 2
is distributed approximately as t with degrees of
freedom given by
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Inference for a Difference in Means of Two
Normal Distributions, Variances Unknown
1. Hypotheses Tests for a Difference in Means,
Variances Unknown
Case 2
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Example 5
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Example 5
Normal probability plot of the arsenic
concentration data from Example 5.
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Example 5
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Example 5
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Minitab Practice for Example 5

• Data file Example 3_5.xls
• Menu ? Stat ? Basic statistics ? 2 sample t ?
  select Samples in different columns ?do not
  check Assume equal variances ?options
• Confident level 95
• Test mean 0.0
• Alternative not equal

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Inference for a Difference in Means of Two
Normal Distributions, Variances Unknown
3. Choice of Sample Size
Two-sided alternative
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Example 6
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Minitab Practice for Example 6

• Output sample size n ?
• Menu ? Stat ? Power and Sample Size ? 2 sample t
  ?
• Different 4
• Power value 0.85
• Sigma 2.7
• Options ? Alternative not equal
• Significant level 0.05

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4. Confidence Interval on the Difference in Means
Case 1
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Example 7
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Example 7
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Example 7
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Example 7
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4. Confidence Interval on the Difference in Means
Case 2
Definition
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Paired t-Test

• A special case of the two-sample t-tests of
  Section 10-3 occurs when the observations on the
  two populations of interest are collected in
  pairs.
• Each pair of observations, say (X1j , X2j ), is
  taken under homogeneous conditions, but these
  conditions may change from one pair to another.
• The test procedure consists of analyzing the
  differences between hardness readings on each
  specimen.

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Paired t-Test
The Paired t-Test
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Example 8
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Example 8
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Example 8
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Paired t-Test
Paired Versus Unpaired Comparisons
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Paired t-Test
A Confidence Interval for ?D
Definition
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Example 9
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Example 9
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Inferences on the Variances of Two Normal
Populations
1. The F Distribution
We wish to test the hypotheses

• The development of a test procedure for these
  hypotheses requires a new probability
  distribution, the F distribution.

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Inferences on the Variances of Two Normal
Populations
1. The F Distribution
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Inferences on the Variances of Two Normal
Populations
1. The F Distribution
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Inferences on the Variances of Two Normal
Populations
1. The F Distribution
The lower-tail percentage points f?-1,u,? can be
found as follows.
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Inferences on the Variances of Two Normal
Populations
2. Hypothesis Tests on the Ratio of Two Variances
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Inferences on the Variances of Two Normal
Populations
3. Hypothesis Tests on the Ratio of Two Variances
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Example 10
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Example 10
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Example 10
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Inferences on the Variances of Two Normal
Populations
4. ?-Error and Choice of Sample Size
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Example 11
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Inferences on the Variances of Two Normal
Populations
5. Confidence Interval on the Ratio of Two
Variances
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Example 12
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Example 12
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Example 12
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Inference on Two Population Proportions
1. Large-Sample Test for H0 p1 p2
We wish to test the hypotheses
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Inference on Two Population Proportions
1. Large-Sample Test for H0 p1 p2
The following test statistic is distributed
approximately as standard normal and is the basis
of the test
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Inference on Two Population Proportions
1. Large-Sample Test for H0 p1 p2
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Example 13
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Example 13
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Example 13
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Minitab Practice for Example 13

• Menu ? Stat ? Basic Statistics ? 2 proportions ?
  Select Summarized Data
• First sample trials 100 successes 27
• Second sample trials 100 successes 19
• Sigma 2.7
• Options ? Alternative not equal
• Confidence level 95
• Test difference 0.0

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Inference on Two Population Proportions
2. ?-Error and Choice of Sample Size
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Inference on Two Population Proportions
2. ?-Error and Choice of Sample Size
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Inference on Two Population Proportions
2. ?-Error and Choice of Sample Size
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Inference on Two Population Proportions
3. Confidence Interval for p1 p2
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Example 14
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Example 14