4 Hypothesis & Testing

CHAPTER OUTLINE 4-1 STATISTICAL INFERENCE 4-2 POINT ESTIMATION 4-3 HYPOTHESIS TESTING Statistical Hypotheses Testing Statistical Hypotheses One-Sided and Two-Sided Hypotheses General Procedure for Hypothesis Testing

4-4 INFERENCE ON THE MEAN OF A POPULATION, VARIANCE KNOWN Hypothesis Testing on the Mean P -Values in Hypothesis Testing Type II Error and Choice of Sample Size Large-Sample Test Some Practical Comments on Hypothesis Testing Confidence Interval on the Mean

4-5 INFERENCE ON THE MEAN OF A POPULATION, VARIANCE UNKNOWN Hypothesis Testing on the Mean P -Value for a t -Test Computer Solution Choice of Sample Size Confidence Interval on the Mean 4-6 INFERENCE ON THE VARIANCE OF A NORMAL POPULATION Hypothesis Testing on the Variance of a Normal Population Confidence Interval on the Variance of a Normal Population

4-7 INFERENCE ON A POPULATION PROPORTION Hypothesis Testing on a Binomial Proportion Type II Error and Choice of Sample Size Confidence Interval on a Binomial Proportion 4-8 SUMMARY TABLE OF INFERENCE PROCEDURES FOR 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 TESTING 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 Testing This 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, H Specify an appropriate alternative hypothesis, H 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 H 0 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 Some Practical Comments on Hypothesis Testing The Eight-Step Procedure 1. Specify the test statistic to be used (such as z 0 ). 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 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 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 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 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