Question

Problem:

The standard deviation of the breaking strengths of certain cables produced by a company is given as 240 lbs. After a change was introduced in the process of manufacture of these cables, the breaking strengths of a sample of 8 cables showed a standard deviation of 300 lbs. investigate the significance of the apparent increase in variability, using a significance level of (a) 0.05 and (b)0.01.

Answer 1

σ₀​=240, N=8, s=300

We assume that the breaking strengths has normal distribution

a)α=0.05

H₀​:σ²=σ₀²​=240²,H₁​:σ²​≠ σ₀²​=240²

x²=((N-1)s²)/σ₀², s² is a random value of corrected variance

This random variable has x²distribution with k=N−1=7 degree of freedom.

Observed value:x²≈10.9375.Critical values: x²_crleft=x²_cr(1-α/2;k)=1.6899

x²_crright=x²_cr(α/2;k)=16.013

Critical region(-∞;1.6899)$\bigcup$ (16.013;∞)

Our observed value does not fall into the critical region.So we accept H₀ - Variability did not change after a change in the process of manufacture.

b)α=0.01

Critical values: x²_crleft=x²_cr(1-α/2;k)=0.9893

x²_crright=x²_cr(α/2;k)=20.278

Critical region(-∞;0.9893)⋃ (20.278;∞)

Our critical value does not fall into the critical region.So we accept H₀ Variability did not change after a change in the process of manufacture.

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