Question

. ​John and Mark participate in a bowling game every week. From past experience, it is known that both scores are normally distributed, where John has a mean score of 145 with a standard deviation of 25, and Mark has a mean score of 160 with a standard deviation of 14. Assuming that their scores are independent, what is the probability that John will have a greater score than Mark in a single game?

Answer 1

pooled std. dev. = sqrt(25^2 + 14^2) = 28.6531

x1bar - x2bar = 145 - 160 = -15

Here, μ = -15, σ = 28.6531 and x = 0. We need to compute P(X >= 0). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (0 - -15)/28.6531 = 0.52

Therefore,
P(X >= 0) = P(z <= (0 - -15)/28.6531)
= P(z >= 0.52)
= 1 - 0.6985
= 0.3015

Ans: 0.3015