Question

1. A new process has been developed for applying photoresist to 125-mm silicon wafers used in manufacturing integrated circuits. Ten wafers were tested, and the following photoresist thickness measurements (in angstroms x1000) were observed:
   13.3987, 13.3957, 13.3902, 13.4015, 13.4001, 13.3918, 13.3965, 13.3925, 13.3946, and 13.4002.
   (a) Test the hypothesis that mean thickness is 13.4 x 1000Å. Use = 0.05 and assume a two sided
   alternative.
   (b) Find a 99% two-sided confidence interval on mean photoresist thickness. Assume that thickness
   is normally distributed.
   (c) Does the normality assumption seem reasonable for these data?

Answer 1

Part a

Here, we have to use one sample t test for the population mean.

H₀: µ = 13.4 versus H_a: µ ≠ 13.4

This is a two tailed test.

From given data, we have

Sample mean = Xbar = 13.39618

Sample standard deviation = S = 0.003908623

Sample size = n = 10

Degrees of freedom = n – 1 = 10 – 1 = 9

Level of significance = α = 0.05

Critical values = -2.2622 and 2.2622

(by using t-table)

Test statistic formula is given as below:

t = (Xbar - µ) / [S/sqrt(n)]

t = (13.39618 – 13.4)/[ 0.003908623/sqrt(10)]

t = (13.39618 – 13.4)/ 0.0012

t = -3.0906

P-value = 0.0129

(by using t-table)

P-value < α = 0.05

So, we reject the null hypothesis

There is insufficient evidence to conclude that the mean thickness is 13.4*1000A.

Part b

Confidence interval for Population mean is given as below:

Confidence interval = Xbar ± t*S/sqrt(n)

WE have

Sample mean = Xbar = 13.39618

Sample standard deviation = S = 0.003908623

Sample size = n = 10

Degrees of freedom = n – 1 = 10 – 1 = 9

Confidence level = 99%

Critical t value = 3.2498

(by using t-table or excel)

Confidence interval = Xbar ± t*S/sqrt(n)

Confidence interval = 13.39618 ± 3.2498*0.003908623/sqrt(10)

Confidence interval = 13.39618 ± 3.2498*0.001236015

Confidence interval = 13.39618 ± 0.0040

Lower limit = 13.39618 - 0.0040 = 13.3922

Upper limit = 13.39618 + 0.0040 = 13.4002

Confidence interval = (13.3922, 13.4002)

Part c

Yes, the normality assumption seem reasonable for these data, because thickness data follows approximately normal distribution.