When a chip fabrication facility is operating normally, the lifetime of a microchip operated at temperature T, measured in degrees Celsius, is given by an exponential (?) random variable X with expected value E[X]=1/? = (200/T) 2 years. Occasionally, the chip fabrication plant has contamination problems and the chips tend to fail much more rapidly. To test for contamination problems, each day m chips are subjected to a one-day test at T=100∘C. Based on?, the number of chips that fail in one day, design a significance test for the null hypothesis H0: the plant is operating normally.

a) Suppose the rejection set of the test is R = {N > 0}. Find the significance level of the test as a function of m, the number of chips tested.

(b) How many chips must be tested so that the significance level is α = 0.01.

(c) If we raise the temperature of the test, does the number of chips we need to test increase or decrease?

Question

Callum Harmon

Mathematics

31 March, 07:15

When a chip fabrication facility is operating normally, the lifetime of a microchip operated at temperature T, measured in degrees Celsius, is given by an exponential (?) random variable X with expected value E[X]=1/? = (200/T) 2 years. Occasionally, the chip fabrication plant has contamination problems and the chips tend to fail much more rapidly. To test for contamination problems, each day m chips are subjected to a one-day test at T=100∘C. Based on?, the number of chips that fail in one day, design a significance test for the null hypothesis H0: the plant is operating normally.

a) Suppose the rejection set of the test is R = {N > 0}. Find the significance level of the test as a function of m, the number of chips tested.

(b) How many chips must be tested so that the significance level is α = 0.01.

(c) If we raise the temperature of the test, does the number of chips we need to test increase or decrease?

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Answers (1)

Know the Answer?

Tristian Nicholson · Answer 1

a. 1 - e^{m/365}

b. 916.60 / approx 917.

c. - 365 T^2 / log (1 - / alpha) / 10000 which is a deceasing function of T.

Step-by-step explanation:

a)

/alpha (m) = P_{H_0} (N>0) = 1 - P_{H_0} (N=0) / / = 1 - p^m.

where p = P_{H_0} (/mbox{The chip survives for 1 day}) / / = P_{H_0} (X> 1 / mbox{day}) = P_{H_0} (X> 1/365 / / mbox{year}) = e^{-/frac{1}{365}}.

Since if T = 100 the lifetime follows exponential with mean = 1 year.

Therefore / alpha (m) = 1 - e^{m/365}

b) At T = 25 the lifetime is exponential with mean = 16 years. Therefore / lambda = / frac{1}{16} and so

/alpha (m) = 1 - e^{/frac{m}{50000}} = 0.01 / Rightarrow m = 916.60 / approx 917.

c) If we raise the temperature during the test the number of chips we need to test (for the same level of significance) will decrease since for a fixed value of / alpha

m = - 365 T^2 / log (1 - / alpha) / 10000 which is a deceasing function of T.