Answer :
Answer:
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
In this problem, we have that:
[tex]\mu = 1.8, \sigma = 1, n = 169, s = \frac{1}{\sqrt{169}} = 0.0769[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{1.9 - 1.8}{0.0769}[/tex]
[tex]Z = 1.3[/tex]
[tex]Z = 1.3[/tex] 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.