STAT1301 Advanced Analysis of Scientific Data - Semester 2, 2019 Workshop 6 - Exercise Sheet Question 1 The first published use of a t-test was in the 1908 paper "The probable error of the mean" by "Student" that appeared in the journal Biometrika. The test analysed data that came from a study to determine whether a new sedative, B, helped increase sleep duration more than an existing sedative, A. In the study each patient was given a tablet of one of the drugs on alternate evenings and the researchers recorded the mean increase in hours of sleep that each drug gave relative to a previous control. The results for the 10 patients are shown in the following table: Patient 1 2 3 4 5 6 7 8 9 10 A 0.7 -1.6 -0.2 -1.2 -0.1 3.4 3.7 0.8 0.0 2.0 B 1.9 0.8 1.1 0.1 -0.1 4.4 5.5 1.6 4.6 3.4 The mean difference between B and A was 1.58 hours with standard deviation 1.23 hours. Student conducted a t-test of the parameter u, the mean difference between B and A in their increase in mean sleep hours, with Ho: u = 0 against H1: u > 0. Carry out this t-test of these hypotheses and give bounds on the P-value. What do you conclude? Re: assumptions of the t-test, i.e. that the data should be close to normally distributed, especially given the small sample size in this case (n=10 differences). Even without constructing a graph (e.g. dotplot, Q-Q plot or boxplot), we can see that the observations are not that far from normally distributed. There is one outlier, but the data distribution is not particularly skewed. Some plots are given below. The outlier looks somewhat worse now, but it is still probably not "extreme". The other 9 observations could easily have come from a normal distribution. Dotplot of differences Normal Q-Q Plot O 4 3 0 Sample Quantiles 2 0 1 - 0 0 0 0 0 O Ā· 0 0 0 1 2 3 4 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Theoretical Quantiles .-
Boxplot of differences 0 0 1 2 3 4 R code to produce these plots: diffs <- c(1.2,2.4,1.3,1.3,0,1,1.8,0.8,4.6,1.4) stripchart(diffs, method = "stack", offset = . 5, at = . 15, pch = 19,main="Dotplot of differences") qqnorm(diffs) boxplot(diffs,main="Boxplot of differences",horizontal=T) We assume that the conditions of the t-test are sufficiently met. The t-statistic for this test is 000 V0 _ 1.58V10 ~ 4.06. 1.23 Using Table 14.2 from the PIDA book, also available at https://people.smp.uq.edu.au/MichaelBulmer/pida/pida52/Tables.pdf we see that, for n-1 = 9 degrees of freedom, 4.06 is between the values of 3.250 and 4.297, which correspond to p-values of 0.005 and 0.001, respectively. We only know that the p-value for our test must be in between these values, so we write 0.001 < p < 0.005 . Note that due to the one-sided alternative hypothesis, we are doing a one-sided hypothesis test here, so there is no need to double the p-value. If you