July, 2000Guang Jin Statistics in Applied Science and Technology Chapter 9_part I ( and 9.7) Tests of Significance

July, 2000Guang Jin Key Concepts in this Section Basis of a Test of Significance Research Questions and Statement of the Hypothesis Test statistics and its distribution Decision Rule Statistical decision and conclusion Z test

July, 2000Guang Jin Basis of a Test of Significance The purpose of a test of significance is to determine whether or not the data provide evidence against the supposition made by the null hypothesis (supposes that there is not effect), in favor of the alternative hypothesis (supposes that there is an effect).

July, 2000Guang Jin Basis of a Test of Significance Test of Significance primarily deal with the problem of “Chance” due to the fact that we usually have to deal with a sample rather than the whole population. In another word, the difference observed is a statistically significant difference (or effect) caused by certain treatment rather than sampling error which is caused by chance.

July, 2000Guang Jin A General Description of Hypothesis Testing Steps Test of Significance such as z-test, t-test (includes one-tailed and two-tailed), ANOVA, etc which usually involves 7 steps: –1. Review of the assumptions: each specific test is only appropriate when its underlying assumption is satisfied. –2. Research Question and Statement of the hypotheses: Research question, the null hypothesis (H 0 ) and alternative hypothesis (H 1 ) should be stated explicitly.

July, 2000Guang Jin A General Description of Hypothesis Testing Steps –3. Test statistic and its distribution: the test statistic (such as z statistic, t statistic or F statistic) is some statistic that may be computed from the data of the sample. The test statistic serves as a decision maker, since the decision to reject or not to reject the H 0 depends on the magnitude of the test statistic. Each statistic has its distribution (such as standard normal distribution for z statistic, t distribution for t statistic and f distribution for F statistic).

July, 2000Guang Jin A General Description of Hypothesis Testing Steps –4. Decision rule: The decision rule tells us to reject the H 0 if the value of the test statistic computed from sample is one of the values in the rejection region and to not reject the H 0 if the computed value of the test statistic is one of the values in the non-rejection region. ( The values of the test statistic forming the rejection region are those values that are less likely to occur if the H 0 is true, while the values making up the acceptance region are more likely to occur if the H 0 is true)

July, 2000Guang Jin A General Description of Hypothesis Testing Steps –5. Calculation of test statistics: from the data contained in the sample we compute a value of the test statistic and compare it with the rejection and non-rejection regions specified in the Decision rule. –6. Statistical Decision: make the decision based on the decision rule.

July, 2000Guang Jin A General Description of Hypothesis Testing Steps –7. Conclusion: if H 0 is rejected, we conclude that H 1 is true. If H 0 is not rejected, we conclude that H 0 may be true. The p value is a number that tells us how unusual our sample results are, given that the null hypothesis is true.

July, 2000Guang Jin Z test for a Population Mean When a hypothesis evaluates how far the observed sample mean deviates, in standard error units, from the hypothesized population mean, it is referred to as a z test, more accurately, as a z test for a population mean.

July, 2000Guang Jin Assumptions of z Test z test is accurate only when –the population is normally distributed or the sample size is large enough to satisfy the requirements of the central limit theorem –the population standard deviation is known –Level of measurement is interval-ratio

July, 2000Guang Jin Applications of z Test (two tailed) 1. Does the ISU students differ from the national average in terms of amount of time they spent on watching TV? (see Example 1 under Schedule)Example 1 2. Does the national average commuting distance describe the mean commuting distance for all workers in the Chicago area? (see Example 2 under Schedule)Example 2

July, 2000Guang Jin Relationship of Test of Significance to Confidence Intervals The decision reached by use of a significance test would be the same as that reached by use of a confidence interval whenever the hypothesis is two-tailed. If a hypothesized difference in means such as  1 -  2 = 0 is included in the CI, H 0 is not rejected. If the hypothesized difference is not included, H 0 is rejected.