AP Statistics Section 11.1 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 ________________ approx. Normal For x : _________________________________ population Normal or CLT (n 30) For p̂ : ________________________ np 10 and n(1 - p) 10 3. _________________________ Independent observations 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 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 estimate 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 2 400 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. 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 provided by the data. 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 : 67 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 So if H 0 is true, and the mean RT this year is still 6.7 minutes, there is about a _____ 1.4% chance that the city manager would obtain a sample of 400 calls with a mean RT of 6.48 minutes or less. The small P-value provides strong evidence against H 0 and in favor of the alternative H a : 67 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 11.1 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 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 A p-value of .2302 indicates that 23.02% of the time we will get a sample where x is at least as big as 17 when 0 . An outcome that would occur this often when 0 is not good evidence that 0. 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 on. 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 .05 but it is not significant level since P < ____ 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 The final step in performing a significance test is to draw a conclusion about the competing claims you were testing. 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: In examples 3 and 4 of this section we simply stated the p-value and interpreted it in the context of the problem. In example 5, we went on to determine if the data was statistically significant be comparing our P-value to our significance level . We can either _______ reject or _______________ 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. Example 6: For the paramedic example, we calculated the P-value to be 0.0139. If we were using an .05 significance level, we would _____ ________ ) reject H : 6.7 minutes ( conclusion since ______ connection ). It appears p .05 ( __________ that the mean response time to all lifethreatening calls this year is less than last year’s average of 6.7 minutes ( context ______ ). 0 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 .05level. But the P-value, 0.0139 gives us a better sense of how strong the evidence against H 0 is. The P-value is the smallest level at which the data are significant.