4Hypothesis & Testing

CHAPTER OUTLINE

4-1 STATISTICAL INFERENCE

4-2 POINT ESTIMATION

4-3 HYPOTHESIS TESTING4-3.1 Statistical Hypotheses4-3.2 Testing Statistical Hypotheses4-3.3 One-Sided and Two-Sided Hypotheses4-3.4 General Procedure for Hypothesis Testing

4-4 INFERENCE ON THE MEAN OF A POPULATION, VARIANCE KNOWN

4-4.1 Hypothesis Testing on the Mean4-4.2 P-Values in Hypothesis Testing4-4.3 Type II Error and Choice of Sample Size4-4.4 Large-Sample Test4-4.5 Some Practical Comments on Hypothesis Testing4-4.6 Confidence Interval on the Mean

4-5 INFERENCE ON THE MEAN OF APOPULATION, VARIANCEUNKNOWN

4-5.1 Hypothesis Testing on the Mean4-5.2 P-Value for a t-Test4-5.3 Computer Solution4-5.4 Choice of Sample Size4-5.5 Confidence Interval on the Mean

4-6 INFERENCE ON THE VARIANCE OF A NORMAL POPULATION

4-6.1 Hypothesis Testing on the Variance of a Normal Population4-6.2 Confidence Interval on the Variance of a Normal Population

4-7 INFERENCE ON A POPULATION PROPORTION4-7.1 Hypothesis Testing on a Binomial Proportion4-7.2 Type II Error and Choice of Sample Size4-7.3 Confidence Interval on a Binomial Proportion

4-8 SUMMARY TABLE OF INFERENCE PROCEDURESFOR A SINGLE SAMPLE

4-9 TESTING FOR GOODNESS OF FIT

4-1 STATISTICAL INFERENCE

• population• sample • parameter estimation• hypothesis testing

4-2 POINT ESTIMATION

•point estimates•point estimator

4-3 HYPOTHESIS TESTING4-3.1 Statistical Hypotheses

•hypothesis•hypothesis testing•comparative experiment

•null hypothesis •alternative hypothesis•two-sided alternative hypothesis•one-sided alternative hypothesis•test of a hypothesis

4-3.2 Testing Statistical Hypotheses

• critical region• acceptance region.• critical values

4-3.3 One-Sided and Two-Sided Hypotheses

• two-sided test• one-sided alternative hypothesis• one-tailed tests

4-3.4 General Procedure for Hypothesis TestingThis chapter develops hypothesis-testing procedures for many practical problems. Use of the following sequence of steps in applying hypothesis-testing methodology is recommended.

1. From the problem context, identify the parameter of interest.

2. State the null hypothesis, H0.

3. Specify an appropriate alternative hypothesis, H1.

4. Choose a significance level .

5. State an appropriate test statistic.

General Procedure for Hypothesis Testing Continued

6. State the rejection region for the statistic.

7. Compute any necessary sample quantities, substitute these into the equation for the test statistic, and compute that value.

8. Decide whether or not H0 should be rejected and report that in the problem context.

Steps 1–4 should be completed prior to examination of the sample data. This sequence of steps will be illustrated in subsequent sections.

4-4 INFERENCE ON THE MEAN OF A POPULATION,VARIANCE KNOWN

• unbiased point estimator

4-4.1 Hypothesis Testing on the Mean

• sampling distribution• test statistic

• acceptance region • critical region or rejection region

4-4.2 P-Values in Hypothesis Testing

• P-value approach

4-4.3 Type II Error and Choice of Sample Size

Finding the Probability of Type II Error

4-4.4 Large-Sample Test

4-4.5 Some Practical Comments on Hypothesis Testing

The Eight-Step Procedure

1. Specify the test statistic to be used (such as z0).2. Specify the location of the critical region (two-tailed, upper-tailed, or lower-tailed).3. Specify the criteria for rejection (typically, the value of , or the P-value at which rejection should occur).

Statistical Versus Practical Significance

• statistical significance• practical significance

The moral of this demonstration is clear: be careful when interpreting the results from hypothesis testing when the sample size is large, because any small departure from the hypothesized value 0 will probably be detected, even when the difference is of little or no practical significance.

4-4.6 Confidence Interval on the Mean

• interval • confidence interval• confidence interval • lower- and upper-confidence limits• confidence coefficient

• two-sided confidence interval• one-sided confidence interval• precision

Relationship Between Tests of Hypotheses and Confidence Intervals

Confidence Level and Precision of Estimation

Choice of Sample Size

One-Sided Confidence Intervals

4-5 INFERENCE ON THE MEAN OF A POPULATION,VARIANCE UNKNOWN

4-5.1 Hypothesis Testing on the Mean

• validity of the assumptions

4-5.2 P-Value for a t-Test

4-5.3 Computer Solution

4-5.4 Choice of Sample Size

• noncentral t distribution • central t distribution• operating curve (or OC) curves

4-5.5 Confidence Interval on the Mean

4-6 INFERENCE ON THE VARIANCE OF A NORMAL POPULATION

4-6.1 Hypothesis Testing on the Variance of a Normal Population

4-6.2 Confidence Interval on the Variance of a Normal Population

One-Sided Confidence Intervals

4-7 INFERENCE ON A POPULATION PROPORTION

4-7.1 Hypothesis Testing on a Binomial Proportion

4-7.2 Type II Error and Choice of Sample Size

4-7.3 Confidence Interval on a Binomial Proportion

• standard error of the point estimator

Choice of Sample Size

One-Sided Confidence Intervals

4-8 SUMMARY TABLE OF INFERENCE PROCEDURES FOR A SINGLE SAMPLE

4-9 TESTING FOR GOODNESS OF FIT

• probability plotting