Question

To test H₀: µ = 42.0 vs. H_A: µ ≠ 42.0, a sample of n = 40 will be taken from a large population with σ= 9.90.

H₀ will be rejected if the sample mean is less than 40.3 or greater than 43.7.

Find and state the level of significance, α, to three (3) places of decimal.

Answer 1

Here, μ = 42, σ = 1.5653, x1 = 40.3 and x2 = 43.7. We need to compute P(40.3<= X <= 43.7). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (40.3 - 42)/1.5653 = -1.09
z2 = (43.7 - 42)/1.5653 = 1.09

Therefore, we get
P(40.3 <= X <= 43.7) = P((43.7 - 42)/1.5653) <= z <= (43.7 - 42)/1.5653)
= P(-1.09 <= z <= 1.09) = P(z <= 1.09) - P(z <= -1.09)
= 0.862 - 0.138
= 0.724

Level of significant = 1 - 0.724 = 0.276