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M 562 Section-10-3

M 562 Section-10-3 - MTH/STA 562 COMMON LARGE-SAMPLE...

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MTH/STA 562 COMMON LARGE-SAMPLE STATISTICAL TESTS In the preceding section, we have laid out a ground work for hypothesis testing which speci°cally includes four basic components ± null hypothesis, alternative hypothesis, test statistic, and rejection region. In practice, the null hypothesis is usually stated so as to specify an exact value of the population parameter ° , namely H 0 : ° = ° 0 , where ° 0 is a speci°ed value of ° . However, there are in general three di/erent kinds of alternative hy- potheses associated with this null hypothesis; they are upper-tail alternative H a : ° > ° 0 , lower-tail alternative H a : ° < ° 0 , and two-tailed alternative H a : ° 6 = ° 0 . In the present section, we shall formally develop the testing procedure with respect to each of the above three alternatives for hypothesis testing on the basis of large samples. For any single population, the sample mean Y and sample proportion b p are known to be respective unbiased estimators of the population mean ± and population proportion p , with standard errors given as ² Y = ² p n and ² b p = r p (1 ° p ) n ; respectively, where ² is the population standard deviation and n is the sample size. Similarly, for the sake of comparing two populations, the di/erence between two sample means, Y 1 ° Y 2 , is an unbiased estimator for the di/erence between two population means, ± 1 ° ± 2 , with standard error ² Y 1 ° Y 2 = s ² 2 1 n 1 + ² 2 2 n 2 ; whereas the di/erence between two sample proportions, b p 1 ° b p 2 , is an unbiased estimator for the di/erence between two population proportions, p 1 ° p 2 , with standard error ² b p 1 ° b p 2 = s p 1 (1 ° p 1 ) n 1 + p 2 (1 ° p 2 ) n 2 ; where ² 2 1 and ² 2 2 are respective population variances and n 1 and n 2 are respective sample sizes. In view of the Central Limit Theorem , it follows that each of the above four unbiased estimators has an approximately normal distribution when samples are large. If the unknown population parameter ° is set to be one of the four population parameters, ± , p , ± 1 ° ± 2 , or p 1 ° p 2 , then for large samples, the random quantity Z = b ° ° ° ² b ° (10.3.1) has approximately a standard normal distribution. 1
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Consider the problem of testing the hypothesis that the unknown population parameter ° is equal to a speci°ed value ° 0 on the basis of a large random sample Y 1 , Y 2 , ± ± ± , Y n . Throughout this section, it is assumed that the unbiased estimator b ° has an approximately normal distribution with mean ° and standard error ² b ° = r V ar ° b ° ± . This is a reasonable assumption as explained in the preceding paragraph. The statistic de°ned in (10 : 3 : 1) will serve as (at least approximately) a test statistic in the following procedures for hypothesis testing on the basis of large samples.
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