Question

The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and standard deviation 1.1 flaws per square yard. This population distribution cannot be normal, because a count takes only whole-number values. An inspector studies 179 square yards of the material, records the number of flaws found in each square yard, and calculates x, the mean number of flaws per square yard inspected. Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds 1.7 per square yard. (Round your answer to four decimal places.

Answer 1

According to the central limit theorem, means of random samples from any distribution will have approximately a normal distribution; approximation gets better with increasing sample size. There is an infinite number of possible means and variances for a continuous random variable so we use “standard normal distribution”

First, you need to find the z-score for 1.7 using the following equation:

Z= (x-mean)/SD;

So Z= (1.7-1.6)/1.1 = 0.0909

Locate this number on your z-score table, and you should find the value of 0.5359. This is the proportion of values that are less than 1.7, so to find the proportion that is greater than 1.7, you need to use 1-0.5359= 0.4641.

So, so the probability that the mean number of flaws exceeds 1.7 would be 0.4641.