Score: Name: Due Wednesday, April 10th in class. β-endorphins are neurotransmitters whose activity has been linked to the reduction of pain in the body. Elite runners often report a runners high during the course of a long distance run. This effect is thought to be due to these compounds. In 1983 Hoaglin, Mosteller, and Tukey published a paper reporting results of a study designed to assess the levels of β-endorphins (fmol/ml) as a response to stress. These investigators selected a group of patients due for surgery and measured their β- endorphin levels 12 hours (low stress) and 10 minutes (high stress) before surgery. Data consistent with the descriptive statistics reported in the reference below appear in Table 1, relevant descriptive statistics are provided in Output 1. Investigators would like to test the hypothesis that β-endorphin levels increase, on average, more than 6 fmol/ml as a result of stress, with a significance level of 0.05. [Hoaglin, D.C., Mosteller, F., Tukey, J.W., (1983). Understanding robust and exploratory data analysis, New York: Wiley] Table 1: β-endorphin levels (fmol/ml) in high and low stress situations Patient 1 2 3 4 5 6 7 8 9 Low 11.5 10.0 7.0 2.0 4.4 5.0 17.0 4.7 5.5 High 9.0 20.0 15.0 2.1 2.5 18.0 42.0 25.0 18.0 Diff -2.5 10.0 8.0 0.1-1.9 13.0 25.0 20.3 12.5 Output 1: Descriptive Statistics for the β-endorphin study - Response = Diff Descriptive Statistics: Diff Variable N Mean SE Mean StDev Minimum Q1 Median Q3 Maximum Diff 9 9.39 3.21 9.64-2.50-0.90 10.00 16.65 25.00 Part 1: Some preliminaries 1.1) The null/alternative hypothesis pair being tested in this setting is: Ho: Ha: Page 1 of 8
1.2) Based on Output 1, do you think that β-endorphin levels increase, on average, more than 6 fmol/ml as a result of stress? Defend your answer. 1.3) Perform the proper hypothesis test using StatCrunch. Report the p-value and give the proper conclusion. In order to do the hypothesis test in StatCrunch: A) Click on Stat. B) Scroll down to T statistics, select One sample and with summary. C) Fill in the sample mean, sample standard deviation, and the sample size using the summary statistics in Output 1, then click Next. D) Change the Null: mean and Alternative to the proper settings. Click Calculate. 1.4) Does your answer from 1.2) match the decision made in 1.3)? Page 2 of 8
Part 2: The power analysis Your answers in 1.2) and 1.3) should not match. However, based on Output 1, it does appear that you should reject Ho. Thus, the power of the test might be too low. Along with setting a significance level, a power analysis should be conducted prior to the onset of a study or the investigators run the risk of missing a significant effect because their sample size is too small or their sample variability is too large. Recall that the power is the probability of rejecting the null hypothesis when it is false. Power = 1 β, where β is the probability of committing a Type II error. Consult the notes from September 19 th for more information. We will be investigating the effects of changing a population s standard deviation, changing the true average of the population represented in Ha, and changing the level of power desired as it pertains to the required sample size. To complete this project you will need the resources at: http://homepage.stat.uiowa.edu/~rlenth/power/ and StatCrunch. Typical power levels required for real life experiments are 0.80 and above. We will look at power from the levels of 0.80, 0.85, and 0.90. Our standard deviation variable will take on values of 2, 5, 8, and 11 and our true (but unknown) population average (which we will call µ Ha ) will take on values of 7, 8, 9, and 10. Note that these are all possible values for µ if the alternative hypothesis is true. So all in all, there are 48 combinations of these 3 variables (3 x 4 x 4) and the applet at the URL listed above will allow us to find the required sample size for each specific combination. Example Table : Required sample sizes for each desired power level at each value of standard deviation. µ Ha = 6.75 Power = 0.80 45 276 705 1331 Power = 0.85 53 321 819 1884 Power = 0.90 62 382 976 2329 the specified standard deviation and the fact that µ Ha = 6.75 Page 3 of 8
2.1) Use the format illustrated in the example table above and complete Tables 2 through 5 A) Pull up http://homepage.stat.uiowa.edu/~rlenth/power/ and select the 1 sample t- test (or paired t) option then click the run selection button at the bottom. Then check to make sure that the significance level is 0.05. B) Adjust the size of the applet box to your liking C) Since our alternative in this case is directional, deselect the two tailed option checkbox at the bottom of the applet D) Above and to the right of each slider is a small square button. Click on that to open a textbox that will allow you to input a specified value instead of trying to use the slider. D.1) Enter the specified value for sigma (standard deviation) D.2) True µ - µ 0. This entry is NOT the specified value of µ Ha but, rather, the difference between the hypothesized average and the value of µ Ha. Statisticians refer to this difference as δ. In general δ = µ µ 0 so for example, the hypothesized average (under Ho) is always 6 for this setting. When µ Ha is 7, δ = µ µ 0 = 7 6 = 1. This is the value that you will H a enter in this box. For the example table I used 0.75 as the input for this box. D.3) Open the Power textbox and enter 0.80 then click on Okay. The required number of observations to achieve the designated power, at the specified standard deviation and difference - δ - between hypothesized and actual averages is output on the line above the power entry. D.4) You will need to reenter the desired power for each iteration. Note that these will not always be exact, and you may have some issues for the larger µ Ha values. Table 2: Required sample sizes for each desired power level at each value of standard deviation. µ Ha = 7 Power = 0.80 H a Power = 0.85 Power = 0.90 the specified standard deviation and the fact that µ Ha = 7 Page 4 of 8
Table 3: Required sample sizes for each desired power level at each value of standard deviation. µ Ha = 8 Power = 0.80 Power = 0.85 Power = 0.90 the specified standard deviation and the fact that µ Ha = 8 Table 4: Required sample sizes for each desired power level at each value of standard deviation. µ Ha = 9 Power = 0.80 Power = 0.85 Power = 0.90 the specified standard deviation and the fact that µ Ha = 7 Page 5 of 8
Table 5: Required sample sizes for each desired power level at each value of standard deviation. µ Ha = 10 Power = 0.80 Power = 0.85 Power = 0.90 the specified standard deviation and the fact that µ Ha = 10 2.2) Within any given µ Ha value, what happens to the required sample size when the desired power level increases? 2.3) Within any given µ Ha value and any given power level, what happens to the required sample size when the dispersion (standard deviation) increases? 2.4) Within any given power level and standard deviation value, what happens to the required sample size as the difference between the hypothesized average and the actual average increases? Page 6 of 8
2.5) Choose any cell in any of your tables. 2.5.1) Interpret the β value for that cell 2.5.2) Interpret the power level for that cell 2.5.3) For all of these tables the significance level α - was set at 0.05. What does this mean? Page 7 of 8
Part 3 The StatCrunch Piece Appropriate graphics can be very helpful in illustrating the concepts you wrote about above. 3.1) Choose any one of the tables you completed. This will be your input data for StatCrunch 3.2) Log onto StatCrunch A) Select the Open StatCrunch B) Enter the results for your chosen table. See Table 6 for an example on the data entry, which uses the data from Table 1. C) Once your data has been entered, click on the graphics button then select Scatter Plot D.1) The x variable is s ; the y variable is n and the group by variable is Power D.2) Click Next and be sure the points and lines boxes are both checked under the display option D.3) Click next then enter an appropriate label for the x and y axis. Then check both boxes for drawing horizontal and vertical gridlines D.4) Click Create Graph to generate a figure similar to Figure 2 below. D.5) Please print your graphic then cut it out and tape/paste it to page 7 underneath your responses to question 2.5.3. Table 6: Example table for generating the comparative power graphic Figure 1: Example graphic for n versus s at three power levels Power s n P.8 2 45 P.8 5 276 P.8 8 705 P.8 11 1331 P.85 2 53 P.85 5 321 P.85 8 819 P.85 11 1884 P.90 2 62 P.90 5 382 P.90 8 976 P.90 11 2329 Page 8 of 8