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AP Statistics Section 9.1C Inference Toolbox for Significance Tests 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 state the hypotheses. You must also define the parameter of interest in the context of the problem. Step 2: Conditions: Choose the appropriate inference procedure and verify the conditions for using it. Step 3: Calculations: Carry out the inference procedure by calculating the test statistic and 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. 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 , x 0 compute the one-sample z-statistic. z s n 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 large n in other cases. approximately correct for __________ 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 wish to test H 0 : 128 H a : 128 Where is the mean systolic blood pressure for the population 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: If we assume the population mean is 128, there is a 27.58% chance of a random sample having a mean systolic blood pressure as extreme as 129.93. There is very little evidence that the mean systolic blood pressure of young execs at this company differs 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 H 0 : 0 vs H a : 0 Where is the mean change in blood pressure of employees. Conditions: SRS: Director chose a random sample of 50 employees. 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: Since our p - value of .017 is less than the significance level of .05, we will reject the H 0 . We conclude that the mean change in blood pressure for the employees at this company is less than 0. We conclude that the mean blood pressure for the employees at this company was lower after the health program then before.