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The number of flaws per square yard in a type of carpet material varies with mean 1.4 flaws per square yard and standard deviation 1.2 flaws per square yard. This population distribution cannot be normal, because a count takes only whole-number values. An inspector studies 178 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 1.5 per square yard.

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According to the Central Limit Theorem, the distribution of the sample means is approximately normal, with the mean equal to the population mean (1.4 flaws per square yard) and standard deviation given by:
[tex]\frac{\sigma}{ \sqrt{n} }=\frac{1.2}{ \sqrt{178} }=0.09[/tex]
The z-score for 1.5 flaws per square yard is:
[tex]z=\frac{1.5-1.4}{0.09}=1.11[/tex]
The cumulative probability for a z-score of 1.11 is 0.8665. Therefore the probability that the mean number of flaws exceeds 1.5 per square yard is
1 - 0.8665 = 0.1335.

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