1. When a production machine is properly calibrated, it requires an average of 26 seconds per unit produced...

Question

1.    When a production machine is properly calibrated, it requires an average of 26 seconds per unit produced, with a standard deviation of 3 seconds. For a simple random sample of n = 36 units, what is the probability that the mean for a simple random sample of this size will be:

a. at least 26.2 seconds?

b. Between 24 seconds and 30 seconds

Answer 1

xbar = sample mean

mu = population mean

sd = standard deviation

n = sample size

a. P(xbar > 26.2)

compute z-stat = (xbar - mu)/(sd/sqrt(n)) = (26.2 - 26)/(3/6) = 0.4

P(xbar > 26.2) = P(z > 0.4) = 1 - P(z < 0.4) = 1 - 0.6554 = 0.3446

b. P(24 < xbar < 30)

compute z-stat for xbar = 24:

(24-26)/(3/6) = -4

compute z-stat for xbar = 30

(30-26)/(3/6) = 8

P(24 < xbar < 30) = P(-4 < z < 8) ~1