Question

An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are independently and normally distributed with a mean of 120 cm and a standard deviation of 4.7 cm.

Find the probability that if 3 are randomly selected, all 3 have lengths that exceed 122 cm.

Answer 1

Use z-score formula:

z = (x - mean)/sd

z = (122 - 120)/4.7 = ?

Once you have the z-score, check a z-table for the probability (keep in mind that you are looking for the probability exceeding 122 cm).

After you have the probability from the table, use a normal approximation to the binomial distribution.

Formulas:
mean = np = 3 * p
sd = √npq = √(3 * p * q)
Note: p = probability from z-table; q = 1 - p

Use z-scores again; this time use 3 for x, the mean calculated above, and the standard deviation calculated above.

Once you have this z-score, determine the probability using a z-table once again.

I hope this will help get you started.