Need help on probability

Question

In a factory, one worker produces an average of 175 units per day with a standard deviation of 20. Another worker produces at an average rate of 165 per day with a standard deviation of 21. What is the probability that the production of worker 1 will be larger than that of worker 2 by more than 400 units during 50 working days?

Answer 1

We can consider the problem to be the probability that the sample mean of worker 1 is more than 400/50 = 8 more than worker 2.

As X1 and X2 are independent, var(X1 - X2) = var(X1) + var(X2) = 20^2 + 21^2 = 841 = 29^2, so the standard deviation of the difference is 29.

Then, for the sample mean, sd(x1-bar - x2-bar) = 29/(sqrt(50)).

Thus, as mean(X1 - X2) = 175 - 165 = 10, P(x1-bar - x2-bar) >= 8 = P(z >= (8-10)/(29/sqrt(50))) =

P(z >= -2 sqrt(50)/29) = P(z >= -0.487659849094171) = 1 - P(z <= -0.487659849094171) =

1 - normsdist(-0.487659849094171) = 0.687104602109854