Question

Two sets of measurements of ethanol concentration in a sample of vodka were made using the same instrument, but on two different days.

On the first day, a standard deviation of s₁ = 9 ppm was found,

and on the next day s₂ = 2 ppm.

Both datasets comprised 6 measurements.

Can the two datasets be combined, or is there is a significant difference at 95% confidence between the datasets, and should we discard one of them?

Hint: Begin by defining the null hypothesis, H₀: s₁² = s₂², and the alternate hypothesis, H_A: s₁² ≠ s₂².

Answer 1

Using the F-test to compare two variances or standard deviations.

When using the F-test, you again require a hypothesis, and have to compare standard deviations. That is, you will test the null hypothesis H₀: σ₁² = σ₂² against an appropriate alternate hypothesis.

Initially starting by defining the null hypothesis, H₀: σ₁² = σ₂²,

whereas alternate hypothesis, H_A: σ₁² ≠ σ₂².

"≠" indicating that this is a 2-tailed test, since interestingly in both cases: σ₁² >σ₂² and σ₁² < σ₂².

Day		Standard Deviation (ppm)
1		9
2		2

A dataset is more reliable if its standard deviation is lower than that of the other dataset. So if we test the null hypothesis H₀: σ₂² = σ₁², against the alternate hypothesis H_A: σ₂² > σ₁².

Since s₂ > s₁, F_calc = s₂²/s₁² = 2²/9² = 4.93*10^-2. The confidence level being 95% we can discard the Day 1 dataset and take Day 2 dataset.