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.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.