A precision engineering firm manufactures silicon wafers. The existing process is specified to have a thickness with a variance of 6 μm26\ \mu m^26 μm2. An engineer suspects that the variance has recently increased due to machine wear.
Ten wafers are randomly selected and their thicknesses are measured, yielding the following results in micrometers:
254,248,251,247,252,250,246,253,255,244 254, 248, 251, 247, 252, 250, 246, 253, 255, 244 254,248,251,247,252,250,246,253,255,244Calculate the sample mean, xˉ\bar{x}xˉ, and the sample variance, s2s^2s2, for these measurements.
Assume that the thicknesses follow a normal distribution.
Test the engineer's suspicion at the 5% significance level, using the hypotheses H0:σ2=6H_0: \sigma^2 = 6H0:σ2=6 and H1:σ2>6H_1: \sigma^2 > 6H1:σ2>6.
A technician introduces a new calibration method which they believe reduces the variance. They provide a sample of 15 wafers where the calculated sample variance is s2=1.95s^2 = 1.95s2=1.95.
Use this value of s2s^2s2 to calculate a 90% confidence interval for the variance, σ2\sigma^2σ2, for wafers produced using the new calibration method.
[You may use P(χ142>6.571)=0.95P(\chi_{14}^2 > 6.571) = 0.95P(χ142>6.571)=0.95 and P(χ142>23.685)=0.05P(\chi_{14}^2 > 23.685) = 0.05P(χ142>23.685)=0.05]
Given the original process variance of σ2=6\sigma^2 = 6σ2=6, evaluate the technician's claim regarding the new calibration method.