## Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found th

Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let X be the random variable representing the time it takes her to complete one review. Assume X is normally distributed. Let X be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews. Complete the distributions.

A. Find the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs.

B. P(_____) = _______.

C. Find the 95th percentile for the mean time to complete one month’s reviews.

D.The 95th Percentile =________.

## Answers ( )

Answer:a)b)The 95th percentile is 4.4935 hours.Step-by-step explanation:To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:Problems of normal distributions can be 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 TheoremThe 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 .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

The reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours.This means that

16 reviews.This means that

A. Find the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs.This is the pvalue of Z when X = 4.25 subtracted by the pvalue of Z when X = 3.5. So

X = 4.25By the Central Limit Theorem

has a pvalue of 0.7967

X = 3.5has a pvalue of 0.0475

0.7967 – 0.0475 = 0.7492

So

C. Find the 95th percentile for the mean time to complete one month’s reviews.This is X when Z has a pvalue of 0.95, so X when Z = 1.645.

The 95th percentile is 4.4935 hours.