The desired percentage of SiO2 in a certain type of aluminous cement is 5.5. To test whether the true average percentage is 5.5
for a particular production facility, 16 independently obtained samples are analyzed. Suppose that the percentage of SiO2 in a sample is normally distributed with and that . a. Does this indicate conclusively that the true average percentage differs from 5.5
This is a test of mean where σ, the population standard deviation is known. The null hypothesis H₀ : µ = µ₀ = 5.5; the alternative H₁ : µ ≠ µ₀. Since σ is known, z = (x − µ₀
)/(σ/√n) is normally distributed for all n and so, for a two-tailed test, we reject H₀ if z ≥ z(α/2) or z ≤ −z(α/2). In this case, α = 0.05 so z(α/2) = z(.025) = 1.960.
P = 2P (Z ≥ |z|) = 2P
*(Z ≥ (x − µ₀)/(
σ/√
n))
⇒ P = 2P
*(Z ≥ (5.25 − 5.5)/(
0.32/√16)) = 2P
*(Z ≥ -3.125) = 2*(0.9991) = 1.9982
We reject the null hypothesis that the true average is 5.5.
