The number of flaws per square yard in a type of carpet material varies with mea

Question

The number of flaws per square yard in a type of carpet material varies with mean 1.3 flaws per square yard and standard deviation 1 flaws per square yard. This population distribution cannot be normal, because a count takes only whole-number values. An inspector studies 161 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 1.4 per square yard. (Round your answer to four decimal places.)

Explanation / Answer

The central limit theorem states that for a sampling distribution (with more than 30 samples) the mean of the sampling distribution is equal to the mean of the population, the standard deviation of sampling distribution (or the standard error) is equal to std.dev of population / sqrt(sample size).Also, the distribution of the sampling mean is approximately normal.



X = 1.4

s.e = 1.1/(161)^(1/2) = 0.086.

if the mean number of flaws exceeds 1.4 per square yard, we convert this into z score

That would be:
z = (1.4-1.83/0.086 = 1.162

Using a z-table the probability of finding more than 1.4 flaws in a sample of 161 yards of carpet is

P ( Z>1.162 )=1P ( Z<1.162 )=10.877=0.123

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