(1) Suppose I have read the data into the data frame naep2013. Typing 3* t . test. {naepZU 13$G4mathchang320 1 1,r naep2013$GBmathchange20 1 1] will perform a 2-sample t-test to compare the mean changes in the Grade 4 math scores from 2011 with the mean changes in the Grade 8 math scores from 2011. The null hypothesis is that the two means are equal (difference in means I 0] and the alternative hypothesis is that the two means are different {difference in means i 0 ]I. (a) Always begin data analysis with an appropriate plot of the data. Start with a side by side boxplot the mean changes in the Grade 4 math scores from 2011 and the mean changes in the Grade 8 math scores from 2011. Include the plot here and comment on it. (b) It's also a good idea to produce summary statistics of your data. Calculate the sample mean and sample variance for each set of math score changes. Include the output here. (c) Now for the hypothesis test - what are the null and alternative hypotheses? (d) Which test will you use, the ttest for independent or dependent samples? What are the assumptions behind the test? Are they satisfied? {e} Run the test and paste the output here. (i) What conclusion do you reach? State the reasoning behind your conclusion and interpret your conclusion. (g) Calculate a 95% condence interval for the difference the mean changes in the Grade 4 math scores from 2011 and the mean changes in the Grade 8 math scores from 2011. Paste the output here. Interpret the confidence interval how does it elaborate upon the conclusion from the hypothesis test above