Question Three: Sample size estimation (8 marks) Wie want to plan a new study, treating the previous trial as a pilot-study under the assumption that the sample estimates are representative for the true population parameters. The pain scores are assumed to be normally distributed. For the new study we want to control a type I error rate of or = 0.05 and maintain a power of at least 1 = 0.8. a. write down the set of hypotheses for the new study, where the estimate of the treatment eect un- der the alternative hypothesis is the difference in means between treatments and controls in the le arthrit 132022 . csv from the previous question. b. Simulate the power (use 1000 simulations) for the t-test and permutation test (B = 1000 permutations) for a sample of 20 [10 treated, 10 controls) patients to reject the null hypothesis. For the simulation, assume the standard deviations for each group are the same as those in arthritis2022 . csv. Explain the results. c. Simulate the power only for the t-test for a range of sample sizes (20 to 200 in treatment group in increments of 20 with an equal number of controls). How many patients do we at least need approximately in our sample to reject the null hypothesis with a probability of 0.8? Copy the relevant R code into your assignment submission. X X AutoSave Off H arthritis2022 v Search (Alt+Q) LI TIAN QI File Page Layout Formulas Data Review View Help Comments Share Home Insert General Conditional Formatting Insert Calibri 11 AA % Format as Table Delete Analyze Sensitivity Paste FE Cell Styles . [Format * Data Clipboard Font Alignment Number Styles Cells Editing Analysis Sensitivity Undo A1 viX fx Control A B C D E F G H J K L M N O P Q R S Control Treatment 64 59 91 73 72 59 56 66 75 60 52 92 56 29 30 31 arthritis2022 + Display Settings + 100% Ready A Accessibility: Unavailable