Question

The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Using Chebyshev's Theorem, bound the probability that the thickness is a) less than 6 or greater than 14 micrometers. b) less than 7 or greater than 15 micrometers (Mo.95.25 p05,15)

Answer 1

Using Chebyshev's Theorem, the probability that the photoresist thickness is less than 6 or greater than 14 micrometers is at least 15/16, and the probability that the thickness is less than 7 or greater than 15 micrometers is at least 24/25.

Step-by-step explanation:

To bound the probability using Chebyshev's Theorem, we will use the formula:

P(μ - kσ < X < μ + kσ) ≥ 1 - 1/k²

(a) For the given case where the thickness is less than 6 or greater than 14 micrometers:

μ - kσ = 10 - k(1) < 6

10 - k < 6

k > 4

μ + kσ = 10 + k(1) > 14

10 + k > 14

k > 4

Since k > 4, 1/k² < 1/16. Therefore, the probability is ≥ 1 - 1/16 = 15/16.

(b) For the case where the thickness is less than 7 or greater than 15 micrometers:

μ - kσ = 10 - k(1) < 7

10 - k < 7

k > 3

μ + kσ = 10 + k(1) > 15

10 + k > 15

k > 5

Since k > 5, 1/k² < 1/25. Therefore, the probability is ≥ 1 - 1/25 = 24/25.