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AP Statistics Section 11.2 A Inference Toolbox for Significance Tests Although the reasoning of significance testing isn’t simple, carrying out a test is. The four-step Inference Toolbox will once again guide us through the inference procedure. Inference Toolbox for Significance Tests Step 1: Hypothesis: Identify the population of interest and the parameter you want to draw conclusions about. State hypotheses. Step 2: Conditions: Choose the appropriate inference procedure. Verify the conditions for using it. Step 3: Calculations: If the conditions are met, carry out the inference procedure by calculating the _______________and test statistic find the ____________. p - value Step 4: Interpretation: Interpret your results in the context of the problem by interpreting the __________ p - value or make a decision about _____ H 0 using statistical significance. Don’t forget the three C’s: ____________, connection and __________. conclusion ____________ context z-Test for a Population Mean To test the hypothesis based on an SRS of size n from a population with unknown mean and known standard deviation , compute the one-sample z-statistic. To determine the p-value, compute the probability of getting a value at least as extreme as the value of our test statistic. The alternative hypothesis ( H a) tells us if we are right-tailed, left-tailed or two-tailed. These P-values are exact if the population is Normal and are approximately correct for __________ large n in other cases. Example 1: The medical director of a large company is concerned about the effects of stress on the company’s younger executives. According to the National Center for Health Statistics, the mean systolic blood pressure for males 35 to 44 years of age is 128, and the standard deviation in this population is 15. The medical director examines the medical records of 72 male executives in this age group and finds that their mean systolic blood pressure is x 129.93. Is this evidence that the mean blood pressure for all the company’s younger male executives is different from the national average? (Assume that the executives have the same as the general population.) Hypothesis : The population of interest is middle - age executives at this company. We want to test the claim the claim that the mean systolic blood pressure for this population is different from the national average of 128. H 0 : 128 H a : 128 Conditions : SRS : Seems safe to assume the 72 were randomly chosen. If not an SRS, results may not generalize to the population. Normality of x : n 72 means the CLT will give x a distribution that is approximately Normal Independence : Since we are sampling without replacement, we must asume that N 10(72) or 720. Calculations : 129.93 - 128 z 1.09 15 72 p value 2(.1379) .2758 Interpretation : A p - value of .2758 indicates that there is a 27.58% chance of a random sample having a mean systolic blood pressure of at least 129.93 when the true population mean is assumed to be 128. There is no good evidence that the mean systolic blood pressure of young execs at this company differ from the national average. Note: The data in Example 1 do not establish that the mean blood pressure for this company’s middle-aged male executives is 128. We simply failed to find convincing evidence that the mean differed from 128. Failing to find evidence against H0 means only that the data are consistent with H0, not that we have clear evidence that H0 is true. Hypothesis : The population of interest is all employees at this company. We want to test a claim that the health promotion campaign caused a drop in the mean blood pressure (before minus after) of employees. H0 : 0 Ha : 0 Conditions : SRS : Director chose a random sample. If not an SRS, results may not generalize to the population. Normality of x : n 50 means the CLT gives a distribution for x that is approx. Normal Independence : Since we are sampling without replacement, we must assume that the population of all employees 10(50) or 500. Calculations : 60 z 2.12 20 50 p value .017 Interpretation: Our p-value of .017 is less than the significance level of .05. We reject the null hypothesis and conclude that the mean difference in blood pressure from before and after the campaign is less than 0. This suggests that the mean blood pressure did decrease in employees at this company.