Food that are high in resistant starch have a low glycemic index (GI) and are beneficial for humanhealth, particularly for those with diabetes. A recent study conducted a randomised double-blindedcontrolled trial to investigate the effectiveness of a type of low-GI rice on the postprandial glucoselevels of diabetic patients, compared to an ordinary “control” rice. Note that “effective” means alower increase in postprandial glucose level compared to the control rice.The 73 diabetic patients who consented to the study were randomly assigned into two groups (GroupsA and B). Their blood glucose was measured immediately prior to and 30 minutes after consuminga prepared meal, comprising low-GI rice for Group A and control rice for Group B. The increase inpostprandial glucose level (i.e., postprandial - preprandial) was recorded. One week later, the sameprocedure was repeated, but with Group A given the control rice and Group B given the low-GI rice.The data is given in the file Rice.csv, available on Blackboard.Handwrite your answers to the following questions. Then scan, photograph or similar, combine intoa single pdf file and upload to the link named Quiz 7 - written questions. Please name your file in theformat: LastName_FirstName/StudentNumber_STAT1301_Quiz7.pdf. If you use R to performcalculations, please append your code at the end of the pdf file.(a) [1 mark] Should the data be analysed as a paired data or two independent samples? Give areason for your answer.(b) [4 marks] Perform an appropriate t-test to test the claim that the low-GI rice is more effectivethan an ordinary rice. Use a 0.05 significance level. Report the null and alternative hypotheses, teststatistic, the P-value, a statistical conclusion, and a conclusion in context.(c) [1 mark] What are the assumptions for the test in part (b)?Consider the following setting of a paired t-test. Let 𝑋𝑖 ∼ 𝑖𝑖𝑑 𝑁(𝜇𝑋, 𝜎𝑋2) denote the “before”measurements and 𝑌𝑖 ∼ 𝑖𝑖𝑑 𝑁(𝜇𝑌, 𝜎𝑌2) denote the “after” measurements, where 𝑖 = 1, 2, … 𝑛.Suppose that Cor(𝑋𝑖, 𝑌𝑖) = 𝜌. Define the difference variables 𝐷𝑖 = 𝑋𝑖 − 𝑌𝑖.(d) [2 marks] Let 𝐷̅ = 1𝑛 ∑ 𝐷𝑖𝑛𝑖=1 . Show that the variance of 𝐷̅ is Var(𝐷̅ ) = 𝜎𝑋2 +𝜎𝑌2−2𝜎𝑋𝜎𝑌𝜌𝑛 .(e) [1 mark] Show that if the samples 𝑋𝑖 and 𝑌𝑖 were independent then Var(𝑋̅ − 𝑌̅ ) = 𝜎𝑋2 +𝜎𝑌2𝑛 .(f) [1 mark] Based on parts (d) and (e), and consider testing 𝐻0: 𝜇𝑋 − 𝜇𝑌 = 0, explain why the pairedt-test is preferred over the two-sample (independent) t-test when 𝜌 > 0.Group Week1 Week2A 3.2 4.6A 1.9 4.2A -0.2 2.1A 2.2 5.2A 4.5 3A 2.9 2.6A 1.9 2.2A 2.9 3.3A 4.2 2.9A 4.1 4.4A 4.5 2.8A 2.3 1.2A 3.8 1.6A 1.9 0.9A 2.7 2.4A 2.6 2.8A 3.4 1A 2.3 1.3A 3.1 4.9A 3.2 4.4A 1.4 3.7A 2.3 1.1A 3.3 3.4A 4.6 3.6A 1.1 2.2A 3.6 3.8A 1 3.3A 2 3.6A 3.6 4.9A 1.3 2.9A 2.6 1.8A 2.6 0.4A 0.5 3A 3.4 4.2A 4.6 5.5A 2.7 4.6A 3.4 4.7B 1.1 3.7B 3.4 3.3B 1.1 2.3B 3 2.9B 3.7 1.4B 4.4 3.2B 4 6.6B 2.8 1.1B 1 3.4B 4.5 3.8B 2.8 2.6B 2.4 2.7B 1.1 4.5B 4.1 3B 1.2 2.3B 4.4 4.1B 2.9 4.2B 3.3 2.9B 5.8 2B 2.6 2.9B 1.7 3.9B 5.2 2.2B 2.1 -0.2B 2 3.4B 5.3 3.7B 5.5 3.1B 3.4 2.5B 0.7 0.1B 1.9 1B 1.8 4.2B 2.7 1.2B 3.2 2.7B 1.4 1B 5 2.3B 3.4 0.6B 4.5 1.4