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10-1 Introduction
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
Figure 10-1 Two independent populations.
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
Assumptions
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
10-2.1 Hypothesis Tests for a Difference in Means, Variances Known
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-1
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-1
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-1
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
10-2.2 Choice of Sample Size
Use of Operating Characteristic Curves
Two-sided alternative:
One-sided alternative:
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
10-2.2 Choice of Sample Size
Sample Size Formulas
Two-sided alternative:
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
10-2.2 Choice of Sample Size
Sample Size Formulas
One-sided alternative:
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-3
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
10-2.3 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.
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
10-2.4 Confidence Interval on a Difference in Means, Variances Known
Definition
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-4
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-4
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
Choice of Sample Size
10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known
One-Sided Confidence Bounds
Upper Confidence Bound
Lower Confidence Bound
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown
We wish to test:
Case 1: 22
221
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown
The pooled estimator of 2:
Case 1: 222
21
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown
Case 1: 222
21
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Definition: The Two-Sample or Pooled t-Test*
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-5
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-5
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-5
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-5
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Minitab Output for Example 10-5
Figure 10-2 Normal probability plot and comparative box plot for the catalyst yield data in Example 10-5. (a) Normal probability plot, (b) Box plots.
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown
22
21 Case 2:
is distributed approximately as t with degrees of freedom given by
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown
22
21 Case 2:
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances UnknownExample 10-6
Example 10-6
Figure 10-3 Normal probability plot of the arsenic concentration data from Example 10-6.
Example 10-6
Example 10-6
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
10-3.3 Choice of Sample Size
Two-sided alternative:
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-7
10-3 Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Minitab Output for Example 10-7
10-3.4 Confidence Interval on the Difference in Means
Case 1:
222
21
10-3.4 Confidence Interval on the Difference in Means
Case 1:
222
21
Example 10-8
10-3.4 Confidence Interval on the Difference in Means
Case 1:
222
21
Example 10-8
10-3.4 Confidence Interval on the Difference in Means
Case 1:
222
21
Example 10-8
10-3.4 Confidence Interval on the Difference in Means
Case 1:
222
21
Example 10-8
10-3.4 Confidence Interval on the Difference in Means
Case 2:
Definition
22
21
• 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.
10-4 Paired t-Test
The Paired t-Test
10-4 Paired t-Test
Example 10-9
10-4 Paired t-Test
Example 10-9
10-4 Paired t-Test
Example 10-9
10-4 Paired t-Test
Paired Versus Unpaired Comparisons
10-4 Paired t-Test
A Confidence Interval for D
10-4 Paired t-Test
Definition
Example 10-10
10-4 Paired t-Test
Example 10-10
10-4 Paired t-Test
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 a new probability distribution, the F distribution.
10-5.1 The F Distribution
10-5 Inferences on the Variances of Two Normal Populations
10-5.1 The F Distribution
10-5 Inferences on the Variances of Two Normal Populations
10-5.1 The F Distribution
10-5 Inferences on the Variances of Two Normal Populations
The lower-tail percentage points f-1,u, can be found as follows.
10-5.3 Hypothesis Tests on the Ratio of Two Variances
10-5 Inferences on the Variances of Two Normal Populations
10-5.3 Hypothesis Tests on the Ratio of Two Variances
10-5 Inferences on the Variances of Two Normal Populations
Example 10-11
10-5 Inferences on the Variances of Two Normal Populations
Example 10-11
10-5 Inferences on the Variances of Two Normal Populations
Example 10-11
10-5 Inferences on the Variances of Two Normal Populations
10-5.4 -Error and Choice of Sample Size
10-5 Inferences on the Variances of Two Normal Populations
Example 10-12
10-5 Inferences on the Variances of Two Normal Populations
10-5.5 Confidence Interval on the Ratio of Two Variances
10-5 Inferences on the Variances of Two Normal Populations
Example 10-13
10-5 Inferences on the Variances of Two Normal Populations
Example 10-13
10-5 Inferences on the Variances of Two Normal Populations
Example 10-13
10-5 Inferences on the Variances of Two Normal Populations
10-6.1 Large-Sample Test for H0: p1 = p2
10-6 Inference on Two Population Proportions
We wish to test the hypotheses:
10-6.1 Large-Sample Test for H0: p1 = p2
10-6 Inference on Two Population Proportions
The following test statistic is distributed approximately as standard normal and is the basis of the test:
10-6.1 Large-Sample Test for H0: p1 = p2
10-6 Inference on Two Population Proportions
Example 10-14
10-6 Inference on Two Population Proportions
Example 10-14
10-6 Inference on Two Population Proportions
Example 10-14
10-6 Inference on Two Population Proportions
Minitab Output for Example 10-14
10-6 Inference on Two Population Proportions
10-6.3 -Error and Choice of Sample Size
10-6 Inference on Two Population Proportions
10-6.3 -Error and Choice of Sample Size
10-6 Inference on Two Population Proportions
10-6.3 -Error and Choice of Sample Size
10-6 Inference on Two Population Proportions
10-6.4 Confidence Interval for p1 – p2
10-6 Inference on Two Population Proportions
Example 10-15
10-6 Inference on Two Population Proportions
Example 10-15
10-6 Inference on Two Population Proportions
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