The number of flaws per square yard in a type of carpet material varies with mean 1.8 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 169 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 1.9 per square yard. (Round your answer to four decimal places.)

0.0968 = 9.68% probability that the mean number of flaws exceeds 1.9 per square yard.

Step-by-step explanation:

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

In this problem, we have that:

Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds 1.9 per square yard.

This is 1 subtracted by the pvalue of Z when X = 1.9. So

By the Central Limit Theorem

has a pvalue of 0.9032

1 - 0.9032 = 0.0968

0.0968 = 9.68% probability that the mean number of flaws exceeds 1.9 per square yard.

step-by-step explanation:

divide, then simplify your answer.

1. make -15.66 ÷   5.8 a fraction. then divide.

2. simplify.  

3. final answer.