H Jay L. Devore Probability and Statistics

3. so the pavement will not be used unless the null hypothesis is rejected

4. 5.

A test with significance level .05 rejects H when a lower-tailed test. 6. With , and , 7. Since , H cannot be rejected. We do not have compelling evi- dence for concluding that ; use of the pavement is not justified. ■ Determination of b and the necessary sample size for these large-sample tests can be based either on specifying a plausible value of s and using the case I formu- las even though s is used in the test or on using the methodology to be introduced shortly in connection with case III. Case III: A Normal Population Distribution When n is small, the Central Limit Theorem CLT can no longer be invoked to jus- tify the use of a large-sample test. We faced this same difficulty in obtaining a small- sample confidence interval CI for m in Chapter 7. Our approach here will be the same one used there: We will assume that the population distribution is at least approximately normal and describe test procedures whose validity rests on this assumption. If an investigator has good reason to believe that the population distri- bution is quite nonnormal, a distribution-free test from Chapter 15 can be used. Alternatively, a statistician can be consulted regarding procedures valid for specific families of population distributions other than the normal family. Or a bootstrap pro- cedure can be developed. The key result on which tests for a normal population mean are based was used in Chapter 7 to derive the one-sample t CI: If is a random sample from a normal distribution, the standardized variable has a t distribution with degrees of freedom df. Consider testing against by using the test statistic . That is, the test statistic results from standardizing under the assumption that H is true using the estimated standard deviation of , rather than . When H is true, the test statistic has a t distribution with df. Knowledge of the test statistic’s dis- tribution when H is true the “null distribution” allows us to construct a rejection region for which the type I error probability is controlled at the desired level. In par- ticular, use of the upper-tail t critical value to specify the rejection region implies that The test statistic is really the same here as in the large-sample case but is la- beled T to emphasize that its null distribution is a t distribution with df rather n 2 1 5 a 5 PT t a ,n21 when T has a t distribution with n 2 1 df Ptype I error 5 PH is rejected when it is true t t a ,n21 t a ,n21 n 2 1 s 1n X S 1n, X T 5 X 2 m S 1n H a : m . m H : m 5 m n 2 1 T 5 X 2 m S 1n X 1 , X 2 , c , X n m , 30 2 .73 . 21.645 z 5 28.76 2 30 12.2647 152 5 2 1.24 1.701 5 2.73 s 5 12.2647 n 5 52, x 5 28.76 z 2 1.645 z 5 x 2 30 s 1n H a : m , 30 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Example 8.9 than the standard normal z distribution. The rejection region for the t test differs from that for the z test only in that a t critical value replaces the z critical value z a . Similar comments apply to alternatives for which a lower-tailed or two-tailed test is appropriate. t a ,n21 The One-Sample t Test Null hypothesis: Test statistic value: Alternative Hypothesis Rejection Region for a Level a Test upper-tailed lower-tailed either or two-tailed t 2t a 2,n21 t t a 2,n21 H a : m 2 m t 2t a , n2 1 H a : m , m t t a , n2 1 H a : m . m t 5 x 2 m s 1n H : m 5 m Glycerol is a major by-product of ethanol fermentation in wine production and con- tributes to the sweetness, body, and fullness of wines. The article “A Rapid and Simple Method for Simultaneous Determination of Glycerol, Fructose, and Glucose in Wine” American J. of Enology and Viticulture, 2007: 279–283 includes the following observations on glycerol concentration mgmL for samples of standard-quality uncertified white wines: 2.67, 4.62, 4.14, 3.81, 3.83. Suppose the desired concentration value is 4. Does the sample data suggest that true average concentration is something other than the desired value? The accompanying normal probability plot from Minitab provides strong support for assuming that the popu- lation distribution of glycerol concentration is normal. Let’s carry out a test of appropriate hypotheses using the one-sample t test with a significance level of .05.