PLEASE SOLVE ALL PARTS (d)|Comparing two means: Consider two measuring instruments that are used to measure the intensity of someelectromagnetic waves. An engineer wants to check if both instruments are calibrated identically, i.e., if they will produceidentical measurements for identical waves. To do so, the engineer does n1 independent measurements of the intensity ofthe a given wave using the first instrument, and n2 measurements on the same wave using the second instrument. Theintegers n, and n2 may not be equal because, for instance, one instrument may be more costly than the other one, or mayproduce measurements more slowly. The measurements are denoted by {X;}"", for the first instrument and by {Y;}"1for the second one. Intrinsic defects of the instruments will lead to measurement errors, and it is reasonable to assumethat {X;}1 are iid Gaussian and so are {Y;}"1- If the two instruments are identically calibrated, {X;}"1 and {Y;}"2,should have the same expectation but may not have the same variance, since the two instruments may not have the sameprecision. Hence, we assume that X; ~ N (µ1,07) and Y; ~ N (µ2, 0), where 41, 42 E R and of,o > 0, and that the twosamples are independent of each other. We want to test whether # = #2. Let i1, fî2, ož, o? be the maximum likelihoodestimators of µ1, 2, of, ož respectively.What is the distribution of+ofLet A = î – jîz. What is the distribution of A?Consider the following hypotheses:Ho : H1 = #2 vs H : µ1# µ2Here and in the next question we assume that of = o3. Based on the previous questions, propose a test withnon-asymptotic level a e (0,1) for Họ against H1.Assume that 10 measurements have been done for both machines. The first instrument measured 8.43 inaverage with sample variance 0.22 and the second instrument 8.07 with sample variance 0.17. Can you conclude thatthe calibrations of the two machines are significantly identical at level 5%? What is (approximately) the p-value ofyour test?

We know that; n1σ^12σ12~χ2n1 and n2σ^22σ22~χ2n2, independently Therefore; n1σ^12σ12+…

(d)|Comparing two means: Consider two measuring instruments that are used to measure the intensity of some electromagnetic waves. An engineer wants to check if both instruments are calibrated identically, i.e., if they will produce identical measurements for identical waves. To do so, the engineer does n independent measurements of the intensity of the a given wave using the first instrument, and n2 measurements on the same wave using the second instrument. The integers n1 and n2 may not be equal because, for instance, one instrument may be more costly than the other one, or may produce measurements more slowly. The measurements are denoted by {X;}""1 for the first instrument and by {Y;}"², for the second one. Intrinsic defects of the instruments will lead to measurement errors, and it is reasonable to assume that {X,} are iid Gaussian and so are {Y;}"1- If the two instruments are identically calibrated, {X,}""1 and {Y;}""I should have the same expectation but may not have the same variance, since the two instruments may not have the same precision. Hence, we assume that X, ~ N (µ1,07) and Y; ~ N (µ2, 03), where µ1, 42 € R and of,ož > 0, and that the two samples are independent of each other. We want to test whether µ1 = µ2. Let î1, fî2, o7, ož be the maximum likelihood estimators of µ1, 42, o, ož respectively. • Vhat is the distribution of + of Let A = jî¡ – fą. What is the distribution of A? Consider the following hypotheses: Họ : µ1 = #2 vs Hị : µ1 # µ2 Here and in the next question we assume that of = o. Based on the previous questions, propose a test with non-asymptotic level a € (0, 1) for Họ against H1. Assume that 10 measurements have been done for both machines. The first instrument measured 8.43 in average with sample variance 0.22 and the second instrument 8.07 with sample variance 0.17. Can you conclude that the calibrations of the two machines are significantly identical at level 5%? What is (approximately) the p-value of your test?