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Carrying Out Significance Tests Review of a Significance Test A test of significance is intended to assess the evidence provided by data against a null hypothesis H0 in favor of an alternate hypothesis Ha. The statement being tested in a test of significance is called the null hypothesis. Usually the null hypothesis is a statement of “no effect” or “no difference.” A one-sided alternate hypothesis exists when we are interested only in deviations from the null hypothesis in one direction H0 : =0 Ha : >0 (or <0) If the problem does not specify the direction of the difference, the alternate hypothesis is two-sided H0: =0 Ha: ≠0 CONDITIONS These should look the same as in the last chapter (for confidence intervals) – SRS – Normality For means—population distribution is Normal or you have a large sample size (n≥30) For proportions--np≥10 and n(1-p)≥10 – Independence CAUTION Be sure to check that the conditions for running a significance test for the population mean are satisfied before you perform any calculations. INFERENCE TOOLBOX (p 705) DO YOU REMEMBER WHAT THE STEPS ARE??? Steps for completing a SIGNIFICANCE TEST: 1—PARAMETER—Identify the population of interest and the parameter you want to draw a conclusion about. STATE YOUR HYPOTHESES! 2—CONDITIONS—Choose the appropriate inference procedure. VERIFY conditions (SRS, Normality, Independence) before using it. 3—CALCULATIONS—If the conditions are met, carry out the inference procedure. 4—INTERPRETATION—Interpret your results in the context of the problem. CONCLUSION, CONNECTION, CONTEXT(meaning that our conclusion about the parameter connects to our work in part 3 and includes appropriate context) Example 1-sided Test The diastolic blood pressure for American women aged 18-44 has approximately the Normal distribution with mean =75 milliliters of mercury (mL Hg) and standard deviation σ=10 mL Hg. We suspect that regular exercise will lower blood pressure. A random sample of 25 women who jog at least five miles a week gives sample mean blood pressure x =71 mL Hg. Is this good evidence that the mean diastolic blood pressure for the population of regular exercisers is lower than 75 mL Hg? Step 1 The parameter of interest is the mean diastolic blood pressure . Our null hypothesis is that the blood pressure is no different for those that exercise. Our alternative hypothesis is one-sided because we suspect that exercisers have lower blood pressure. H0: = 75 Ha: < 75 Step 2 Since we know the population standard deviation we will be performing a z-test of significance. We were told that the sample is random, but we do not know if it is an SRS from the population of interest. This may limit our ability to generalize. Since the population distribution is approximately Normal, we know that the sampling distribution of x will also be approximately Normal. So we are safe using the z procedures. The blood pressure measurements for the 25 joggers should be independent. Note that the population of interest is at least 10 times as large as the sample. Step 3 A curve should be drawn, labeled, and shaded. You can use the formula to calculate your z test statistic for this problem x 0 z In this case z = -2.00 n Mark this on your sketch. Based on our calculations the P-value is 0.0228. x 71 , σ=10, n=25 Step 4 Since there is no predetermined level of significance if we are seeking to make a decision, this could be argued either way. If exercisers are no different, we would get results this small or smaller about 2.28% of the time by chance. This result is significant at the 5% level, but is not signficant at the 1% level. We would likely reject H0. There is not much chance of obtaining a sample like we did if there is no difference, so we would reject the idea that there is no difference and conclude that the mean diastolic blood pressure of American women aged 18-44 that exercise regularly is probably less than 75 mL Hg. DUALITY A level α two-sided significance test rejects a hypothesis H0 : = 0 exactly when 0 falls outside a level 1- α confidence interval for . This relationship is EXACT for a TWO-SIDED hypothesis test FOR A MEAN, but IS NOT EXACT FOR tests involving PROPORTIONS. Essentially, if the parameter value given in the null hypothesis falls inside the confidence interval, then that value is plausible. If the parameter value lands outside the confidence interval, then we have good reason to doubt H0.