Question

A lapping process which is used to grind certain silicon wafers to proper thickness (100 mil). The sample of 20 has produced the standard deviation of 0.50 mil. Use α= 0.10 level of significance to establish the confidence level of variance (Select both upper and lower limits).

Answer 1

Solution:

Here, we have to find the confidence interval for the population standard deviation.

Confidence interval for population standard deviation is given as below:

Sqrt[(n – 1)*S² / χ²_{α/2, n – 1} ] < σ < sqrt[(n – 1)*S² / χ² _{1 - α/2, n – 1} ]

We are given

Significance level = α = 0.10 = 10%

Confidence level = 1 - α = 1 - 0.10 = 0.90 = 90%

Sample size = n = 20

Degrees of freedom = n – 1 = 19

Sample standard deviation = S = 0.50

Upper critical value = χ² _R = χ²_{α/2, n – 1} = 30.1435

Lower critical value = χ² _L = χ² _{1 - α/2, n – 1} = 10.1170

(By using chi square table)

Sqrt[(n – 1)*S² / χ²_{α/2, n – 1} ] < σ < sqrt[(n – 1)*S² / χ² _{1 - α/2, n – 1} ]

Sqrt[(20 – 1)*0.50^2 / 30.1435] < σ < sqrt[(20 – 1)*0.50^2 / 10.1170]

Sqrt(0.1576) < σ < Sqrt(0.4695)

0.3970 < σ < 0.6852

Lower limit = 0.3970

Upper limit = 0.6852