Problem 2 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 ni independent measurements of the intensity of 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 X1,..., Xn for the first instrument and by Y1,..., Yn, for the second one. Intrinsic defects of the instruments will lead to measurement errors, and it is reasonable to assume that X1,..., X are iid Gaussian and so are Yı, ..., Yng. If the two instruments are identically calibrated, the X,'s and the Y,'s should have the same expectation but may not have the same variance, since the two instruments may not have the same precision. i.i.d. i.i.d. N(H1, o}) and Yı,..., Yna "* N(H2, 03), Hence, we assume that X1, .., Xn where u1, l2 ER and o, o> 0, and that the two samples are independent of each other. We want to test whether ui = µ2. 1. Recall the expression of the maximum likelihood estimators for (u1, o) and for (42, 0;). Denote these estimators by (@1, ô?) and (2, ô;). and of of Recall the distribution of What is the distribution of of Let A = 1 - în. What is the distribution of A ? %3D Consider the following hypotheses: Ho : "H1 = 42" vs. Ho : " H1 = H2". Here and in the next question we assume that of = o3. Based on the previous questions, propose a test with non asymptotic level a E (0, 1) for Ho against H1. %3D Assume that 10 measurements have been done for both machines. The first in- strument measured 8.43 in average with sample variance 0.22 and the second in- strument measured 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 ?
Problem 2 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 ni independent measurements of the intensity of 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 X1,..., Xn for the first instrument and by Y1,..., Yn, for the second one. Intrinsic defects of the instruments will lead to measurement errors, and it is reasonable to assume that X1,..., X are iid Gaussian and so are Yı, ..., Yng. If the two instruments are identically calibrated, the X,'s and the Y,'s should have the same expectation but may not have the same variance, since the two instruments may not have the same precision. i.i.d. i.i.d. N(H1, o}) and Yı,..., Yna "* N(H2, 03), Hence, we assume that X1, .., Xn where u1, l2 ER and o, o> 0, and that the two samples are independent of each other. We want to test whether ui = µ2. 1. Recall the expression of the maximum likelihood estimators for (u1, o) and for (42, 0;). Denote these estimators by (@1, ô?) and (2, ô;). and of of Recall the distribution of What is the distribution of of Let A = 1 - în. What is the distribution of A ? %3D Consider the following hypotheses: Ho : "H1 = 42" vs. Ho : " H1 = H2". Here and in the next question we assume that of = o3. Based on the previous questions, propose a test with non asymptotic level a E (0, 1) for Ho against H1. %3D Assume that 10 measurements have been done for both machines. The first in- strument measured 8.43 in average with sample variance 0.22 and the second in- strument measured 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 ?
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Transcribed Image Text:Problem 2
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 ni independent measurements of the intensity of a given wave using the
first instrument, and n2 measurements on the same wave using the second instrument.
The integers ni 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 X1, ..., Xn, for the first instrument and by Y1,..., Yn, for
the second one. Intrinsic defects of the instruments will lead to measurement errors, and
it is reasonable to assume that X1,..., Xn are iid Gaussian and so are Y1,... , Yn.
the two instruments are identically calibrated, the X;'s and the Y;'s should have the same
expectation but may not have the same variance, since the two instruments may not have
the same precision.
If
i.i.d.
N(42, 03),
i.i.d.
N(41, 0?) and Y1,... , Yn2
Hence, we assume that X1, ..., Xn1
where u1, µ2 ER and o?, o, > 0, and that the two samples are independent of each other.
We want to test whether µ1 = µ2.
1. Recall the expression of the maximum likelihood estimators for (u1,0?) and for
(u2, o). Denote these estimators by (u1, ô?) and (u2, ô;).
and of
of
2. Recall the distribution of
3. What is the distribution of
+
4. Let A = û1 - 2. What is the distribution of A ?
5. Consider the following hypotheses:
Ho : "H1 = H2" vs. Ho : "1 = H2".
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 Ho against H1.
6. Assume that 10 measurements have been done for both machines. The first in-
strument measured 8.43 in average with sample variance 0.22 and the second in-
strument measured 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 ?
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