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Using Power to Determine Sample Size whether the diet served at the college cafeteria is causing excess weight gain. The effect (lbs) is the mean weight gain (lbs) of first-year student residents after one semester of living in a dormitory. To determine the number of students that ought to be included in the sample, Power We are planning a study to determine 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 the following power-plot was generated. 4 6 Effect 8 10 12 1. What null and alternative hypotheses are being tested? a. H0: effect = 0 vs. HA: effect > 0 b. H0: effect ≤ 0 vs. HA: effect > 0 c. Choices (a) and (b) are equivalent, and both are “correct”. Choice (a) emphasizes that 0 is the hypothetical value of the parameter effect to which the observed value will be compared in the test, i.e., in the test statistic and in calculation of the P-value. 2. The sample sizes used in the graph Label each of the four curves with their sample size. Which curve is based on the least sample size of 32? a. orange b. blue c. green d. red e. not enough information is given Power are 32, 64, 128, and 256 students. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Effect 8 10 12 3. Which of the four sample sizes would be optimal for detecting an acceptable power is 0.90 (i.e., 90%)? a. 32 (red) b. 64 (green) c. 128 (blue) d. 256 (orange) Power effect of 5 lbs, if the minimum 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Effect 8 10 12 For detecting an effect of 5 lbs, if the minimum acceptable power is 0.90 (i.e., 90%), why would a sample size of 32 be too small a. Power would be 0.45, too small, smaller than 0.90 b. Power would be 0.55, too small, smaller than 0.15 c. Both of the above d. None of the above Power 4. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Effect 8 10 12 For detecting an effect of 5 lbs, if the minimum acceptable power is 0.90 (i.e., 90%), why would a sample size of 256 be too large? a. A sample of size 128 (blue) would be sufficient, with power = 0.93 and Type 2 error rate = 0.17 b. A sample of size 256 would be wasting resources, because the power would be greater than required. c. Both of the above d. None of the above Power 5. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Effect 8 10 12 6. Which of the four sample sizes would be optimal for detecting an acceptable Type 2 error rate is 0.15 (i.e., 15%)? a. 32 (red) b. 64 (green) c. 128 (blue) d. 256 (orange) Power effect of 6 lbs, if the maximum 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Effect 8 10 12 Would increasing the significance level from 0.05 to 0.10 increase, decrease, or not change the Type 2 error rate? a. increase b. decrease c. not change Power 7. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Effect 8 10 12 Would decreasing the Type 1 error rate from 0.05 to 0.01 increase, decrease, or not change the Type 2 error rate? a. increase b. decrease c. not change Power 8. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Effect 8 10 12 We could redesign the study by observing only one sex, male or female. While this would limit the scope of inference to a narrower population, it would decrease the standard deviation of the underlying random variable of interest, weight gain (lbs) = effect (lbs). Would decreasing the standard deviation increase, decrease, or not change the Type 2 error rate? a. increase b. decrease c. not change Power 9. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Effect 8 10 12 10. Would changing the Type 2 error rate (by changing the sample size, and/or the standard deviation) change the Type 1 error rate? a. Yes b. No, the Type 1 error rate is fixed at the set value of the significance level. Power the minimum effect to detect, 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Effect 8 10 12