Question 1 [50] The dataset InsectSpray in the datasets package in R contains data that were collected on the counts of insects in agricultural experimental units treated with 6 different insecticides. The purpose of the study was to investigate the effectiveness of the sprays on the insects. Use R-software to answer the questions that follow. Present or attach the R codes and outputs used to answer the questions. 1.1 Use a boxplot to represent the data. Interpret the output. (6) 1.2 Perform an analysis of variance and present your results on an ANOVA table. Test the hy- pothesis that there are no differences in the mean effectiveness of the sprays using a 5% level of significance. For the test provide the null and alternative hypotheses, critical region (or rejection rejection), test statistic and conclusion. (12) 1.3 Using Tukey's honest significant difference (HSD), investigate the pairwise differences among the six sprays. Are there any significant pairwise differences at 5% level of significance? For the tests, provide the null and alternative hypotheses, critical regions (or rejection rejections), test statistics and conclusions. (18) 1.4 For the conclusions in parts 1.2 and 1.3 to be valid, the assumptions of normality, ho- moscedasticity and independence must be met by the data. Use the residuals from the fitted model to examine the three conditions. (a) Was there a significant evidence of a violation of the normality condition? Explain. (5) (b) Was there a significant evidence that the variance in the insect counts was different for all the six sprays? Discuss. (5) (c) Is the assumption of independent residuals plausible? Justify your answer. (4)