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AP Statistics Section 9.1 B More on Significance Tests Conditions for Significance Tests The three conditions that should be satisfied before we conduct a hypothesis test about an unknown population mean or proportion are the same as they were for confidence intervals: 1. _______ SRS from the population of interest. 2. Distribution of x and p̂ must be approximately Normal x For : _________________________________ population Normal or CLT (n 30) p̂ For : ________________________ np 10 and n(1 - p) 10 Independent observations 3. _________________________ If sampling w/o replacement ___________ N 10n Example 1: Check that the conditions from the paramedic example in section 11.1 A are met. SRS: SRS of 400 Normality of x : n 400 so CLT gives x a dist. that is approx. Normal Independence: Calls without replacement so pop. of all calls 10(400) or 4000 Test Statistics A significance test uses data in the form of a test statistic. The following principles apply to most tests: (1) the test statistic compares the value of the parameter as stated in the H __0 to an estimate of the parameter from the sample data. (2) values of the sample statistic far from the parameter value in the direction specified by the alternative hypothesis give evidence _____________ against H 0 (3) to assess how far the estimate is from the parameter, standardize the estimate. In many common situations, the test statistic has the form: sample value - hypothesized value test statistic = ----------------------------------------standard deviation of the sample dist. 6.48 6.7 z 2.2 2 400 Why z ? We know the population standard deviation. Because the result is over two standard deviations below the hypothesized mean 6.7, it gives good evidence that the mean RT this year is not equal to 6.7 minutes, but rather, less than 6.7 minutes. P-values The probability, computed assuming __________, H 0 is true that the observed sample outcome would take a value as extreme as or more extreme than that actually observed is called the __________ p - value of the test. The smaller the P-value is, the stronger the evidence is against H 0 provided by the data. I suggest the following format when interpreting a pvalue: parameter is ________, Assuming the population ________ null value there is a ________________% P value as a percent chance of getting a sample statistic as extreme as ___________. sample statistic This is little / mod erate / strong evidence that _________________ _____________________________. give a conclusion about the H a in context Example 3: Let’s go back to our paramedic example. The P-value is the probability of getting a sample result at least as extreme as the one we did ( x = 6.48) if H 0 : 6.7 were true. In other words, the P-value is P( x 6.48) calculated assuming 6.7 . We just found the z-score for this exact situation, so using Table A or our calculator, this P-value is _______. .0139 Interpret this p-value. normalcdf (10000,2.2,0,1) Assuming the population mean is 6.7, there is a 1.39% chance of getting a sample statistic as extreme as 6.48. This is strong evidence that the mean response time is less than 6.7 minutes. If the Ha is two-sided, both directions count when figuring the P-value. Example 4: Suppose we know that differences in job satisfaction scores in Example 3 of section 9.1 A follow a Normal distribution with standard deviation 60. If there is no difference in job satisfaction between the two work environments, the mean is _______. 0 Thus H0: ________. 0 The Ha says simply “there is a difference,” thus Ha:________. 0 Data from 18 workers gave x 17. That is, these workers preferred the self-paced environment on average. Find the p-value for this situation and interpret it. 17 0 z 1.20 60 18 .1151 P - value 2(.1151) .2302 Assuming the mean of the population differences of JDS scores is 0, there is a 23.02% chance of getting a sample mean difference as extreme as 17. This is very little evidence of a difference between job satisfaction for machine paced vs self paced. Statistical Significance We can compare the P-value with a fixed value that we regard as decisive. This amounts to announcing in advance how much evidence against H 0 we will insist upon. The decisive value of P is called the significance level. We write it as ____, the Greek letter alpha. If the P-value , we say that the data are statistically significant at level Example 5: Back to the paramedic example. We found the P = 0.0139. The result is statistically significant at the .05 level since P < .05 but it is not significant at the .01 level since P > .01 “Significant” in the statistical sense does not mean “_____________.” important It means simply “not likely to happen just by chance _________.” Interpreting Results in Context As with confidence intervals, your conclusion should have a clear connection to your calculations and should be stated in the context of the problem. These are called the 3 C’s. In significance testing, there are two accepted methods for drawing conclusions. If no significance level is given, you will simply interpret the p-value in context as we just did in examples 3 & 4. If a significance level is given, we can either _______or reject ______________ fail to reject the Ho based on whether our result is statistically significant at a given significance level. Warning: if you are going to draw a conclusion based on statistical significance, then the significance level should be stated before the data are produced. I suggest the following formats when writing your conclusion to a test with a significance level: Example 6: Consider our paramedic example again. If we conducted a .05 significance test, write a conclusion in context. Since the P - value of .0139 is less than our significance level of .05, I will reject the H o in favor of the H a . I can conclude that the mean of the paramedic response times is less than 6.7 minutes. Example 7: Consider our paramedic example once again. If we conducted a .01 significance test, write a conclusion in context. Since the P - value of .0139 is greater than our significance level of .01, I fail to reject the H o . There is insufficient evidence to conclude that the true mean of the paramedic response times is less then 6.7 minutes. Finally, stating a P-value is more informative than simply giving a “reject” or “fail to reject” conclusion at a given significance level. For example, a P-value of 0.0139 allows us to reject H 0 at the .05 level. But the P-value, 0.0139 gives us a better sense of how strong the evidence against H 0 is.