(b) Calculate a 95% confidence interval forˆβ1. From the result of CI, what conclusion can you draw aboutthe relationship betweenxandY? Is it consistent with the scatter plot?

(c) Predict the value ofYatx*= 11 by using the regression model. Find 95% prediction intervalYatx*= 11.

(d) Which prediction interval will be wider, the one atx*= 5.9 or the one atx*= 14.6? Explain withoutcalculations.5

6. Suez et al.(Nature2014, 514:181) studied the effect of artificial sweeteners on blood glucose regulation.In one experiment, they fed mice with a normal diet, supplemented in their drinks either by glucose or bysaccharin (a non-caloric artificial sweetener). Mice were randomly selected to receive either the glucose orthe saccharin treatment. After 22 weeks, the mice were subject to a glucose tolerance test to measure theirability to regulate glucose in their blood.A high glycemic response means poor regulation (high glucoseconcentrations). Part of their glycemic response data is given below for 11 mice in each treatment.glucose22.113.616.317.614.316.617.018.815.214.915.6saccharin21.822.316.831.220.619.327.420.923.222.118.6(a) State (briefly) the assumptions you must make to proceed with an analysis of this problem. Explainwhy an equal variance assumption is reasonable (You do not need to assess the validity of the otherassumptions for this question.)(b) Using the data above, perform a hypothesis test of the claim that mice under the two supplement treat-ments (glucose and saccharin) have the same mean glycemic response, versus the two-sided alternative.Here the two samples are independent, not paired. Let{X1, . . . , Xn1}denotes the glycemic response (inmg/dl) of the mice that received glucose treatment, and the sample given in this problem is{22.1,13.6,16.3,17.6,14.3,16.6,17.0,18.8,15.2,14.9,15.6}. Let{Y1, . . . , Yn2}denotes the glycemic response (in mg/dl)of the mice that received saccharin treatment, and the sample given in this problem is{21.8,22.3,16.8,31.2,20.6,1,27.4,20.9,23.2,22.1,18.6}.i. In order to do the two-sample T-test, the assumptions should be satisfied first that{X1, . . . , Xn1}isa random sample fromN(μ1, σ21), and{Y1, . . . , Yn2}is a random sample fromN(μ2, σ22),{X1, . . . , Xn1}and{Y1, . . . , Yn2}are independent of each other.ii.H0:μ1-μ2= 0 versusH1:μ1-μ26= 0.•From the data,n1=n2= 11, our observed ¯x= 16.55, ¯y= 22.20,s21= 5.64,s22