Question

Achievement test scores of all high school seniors in a state have mean 60 and variance 64. A random sample of n = 100 students from one large high school had a mean score of 58. Is there evidence to suggest that this high school is inferior? (Calculate the probability that the sample mean is at most 58 when n = 100.) Show the appropriate graph of the sample mean.

Answer 1

Here, μ = 60, σ = sqrt(64/100) = 0.8 and x = 58. We need to compute P(X <= 58). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (58 - 60)/0.8 = -2.5

Therefore,
P(X <= 58) = P(z <= (58 - 60)/0.8)
= P(z <= -2.5)
= 0.0062