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 1.7 flaws per square yard and standard deviation 0.8 flaws per square yard. This population distribution cannot be normal, because a count takes only whole-number values. An inspector studies 160 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 1.8per square yard. (Round your answer to four decimal places.)

Explanation / Answer

Ans:

Using the central limit theorem here requires that we consider the 160 yard sample the inspector choose to be a random sample of 160 individual 1 yard samples.

Now the central limit theorem say 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), and the the distribution of the sampling means is approximately normal.

Now putting this all together we get the following:

mean = 1.7

s.e =0.8/sqrt(160) = 0.0632
Now to see if the mean number of flaws exceeds 1.8 per square yard, we find a Z-score (for the sampling distribution) for the value 1.8.

That would be:
z = (1.8-1.7)/0.0632 = 1.58

P(z>1.58)=1-P(z<1.58)=1-0.9429=0.0571

The probability of finding more than 1.8 flaws in a sample of 160 yards of carpet is 0.0571

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