Question 2 [17 marks] A primary school organised a "paper airplane'I morning where students spent the morning designing and making paper airpianes. In the afternoon, the students then tested their paper airplanes to see how far they would fly. The teachers organised a friendly contest to compare paper airplanes made by students from different grades. A random sample of six students was seiected from each of grades 1, 2, 3 and 4 and for each student. their paper airplane's flying distance {in metres] was recorded. Some sample statistics are displayed in the table below and the teachers decided to perform some analyses based on these data. For each grade, assume the paper airplane flying distances are normaily distributed. Grade Sum of Sample Values Sum of SquaIed Sample Values 2 EL Y2.- = 31.2 25;, Y? = 133.33 3 23:1 Y3.- = 35.1 2;, Y3? = 293.09 4 EL] Y4.- = 36.9 3;, Y3 = 229.69 A. [2 marks] If the teachers assume that the population variance of the paper airplane fiying distances is known and equal to 0-2 = 0135 for ali grades, calculate a 99% confidence intervai for the difference in true mean paper airplane flying distance between grades 1 and 4. For parts B and C, the teachers now assume that the population variances of the paper airplane flying distances are unknown for all grades. B. [4 marks] Test whether the population variance of paper airplane flying distances is the same for grades 1 and 4. Clearly state your hypotheses and use a significance level of 01 = 1%. C. [2 marks] Caicuiate a 99% confidence interval for the difference in true mean paper airplane flying distance between grades 1 and 4. D. [4 marks] Comparing the confidence intervals catculated in parts A and C, explain the conclusion that is implied by each confidence interval. Also, provide a statistical explanation for the differences between the confidence intervals. The teachers now assume that the population variances of the paper airplane flying distances are still unknown but equal for all four grades and decide to perform a one-way ANOVA on this data with grade as the factor. E. [2 marks] Caiculate the sum of squares for treatment for the one-way ANOVA. F. [3 marks] Test whether there is a difference in the true mean paper airplane fiying distance between the four grades. Cleariy state your hypotheses and use a significance Level of a = 1%