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Hypothesis & Testing

This chapter outlines the process of statistical inference, including hypothesis testing and point estimation. It covers statistical hypotheses, testing procedures, one-sided and two-sided hypotheses, and the general procedure for hypothesis testing. It also discusses inference on the mean and variance of a population, inference on a population proportion, and testing for goodness of fit.

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Hypothesis & Testing

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  1. 4 Hypothesis & Testing

  2. CHAPTER OUTLINE • 4-1 STATISTICAL INFERENCE • 4-2 POINT ESTIMATION • 4-3 HYPOTHESIS TESTING • 4-3.1 Statistical Hypotheses • 4-3.2 Testing Statistical Hypotheses • 4-3.3 One-Sided and Two-Sided Hypotheses • 4-3.4 General Procedure for Hypothesis Testing

  3. 4-4 INFERENCE ON THE MEAN OF A POPULATION, VARIANCE KNOWN • 4-4.1 Hypothesis Testing on the Mean • 4-4.2 P-Values in Hypothesis Testing • 4-4.3 Type II Error and Choice of Sample Size • 4-4.4 Large-Sample Test • 4-4.5 Some Practical Comments on Hypothesis Testing • 4-4.6 Confidence Interval on the Mean

  4. 4-5 INFERENCE ON THE MEAN OF A • POPULATION, VARIANCE • UNKNOWN • 4-5.1 Hypothesis Testing on the Mean • 4-5.2 P-Value for a t-Test • 4-5.3 Computer Solution • 4-5.4 Choice of Sample Size • 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

  5. 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 • 4-8 SUMMARY TABLE OF INFERENCE PROCEDURES • FOR A SINGLE SAMPLE • 4-9 TESTING FOR GOODNESS OF FIT

  6. 4-1 STATISTICAL INFERENCE • population • sample • parameter estimation • hypothesis testing

  7. 4-2 POINT ESTIMATION • point estimates • point estimator

  8. 4-3 HYPOTHESIS TESTING • 4-3.1 Statistical Hypotheses • hypothesis • hypothesis testing • comparative experiment

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

  10. 4-3.2 Testing Statistical Hypotheses • critical region • acceptance region. • critical values

  11. 4-3.3 One-Sided and Two-Sided Hypotheses • two-sided test • one-sided alternative hypothesis • one-tailed tests

  12. 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, H0. 3. Specify an appropriate alternative hypothesis, H1. 4. Choose a significance level a. 5. State an appropriate test statistic.

  13. 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.

  14. 4-4 INFERENCE ON THE MEAN OF A POPULATION, • VARIANCE KNOWN • unbiased point estimator

  15. 4-4.1 Hypothesis Testing on the Mean • sampling distribution • test statistic

  16. acceptance region • critical region or rejection region

  17. 4-4.2 P-Values in Hypothesis Testing • P-value approach

  18. 4-4.3 Type II Error and Choice of Sample Size Finding the Probability of Type II Error b

  19. 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 a, or the P-value at which rejection should occur).

  20. Statistical Versus Practical Significance • statistical significance • practical significance

  21. 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 m0 will probably be detected, even when the difference is of little or no practical significance.

  22. 4-4.6 Confidence Interval on the Mean • interval • confidence interval • confidence interval • lower- and upper-confidence limits • confidence coefficient

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

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