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Expert-verified Found in: Page 427 ### Introductory Statistics

Book edition OER 2018
Author(s) Barbara Illowsky, Susan Dean
Pages 902 pages
ISBN 9781938168208 # 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 Χ be the random variable representing the time it takes her to complete one review. Assume Χ is normally distributed. Let x- be the random variable representing the meantime to complete the $16$ reviews. Assume that the $16$ reviews represent a random set of reviews. What causes the probabilities in Exercise $7.3$andExercise $7.4$to be different?

Here, X and $\stackrel{-}{x}$ are two different variables, and the distribution of X and $\stackrel{-}{x}$ have the same means but different standard deviations. The distribution of X is more spread than $\stackrel{-}{x}$. This causes the probabilities, P($3.5$<x<$4.25$) and P($3.5$<$\stackrel{-}{x}$<$4.25$) to be different.

See the step by step solution

## Step 1: Given Information

The required probability in Exercise $7.3$ is P($3.5$<x<$4.25$) and in Exercise $7.4$ is P($3.5$<$\stackrel{-}{x}$<$4.25$).

## Step 2: Explanation

We have to evaluate the probability that required to be computed in Exercise $7.3$ and in Exercise $7.4$.

The required probability in Exercise $7.3$ is P($3.5$<x<$4.25$) and in Exercise $7.4$ is P($3.5$<$\stackrel{-}{x}$<$4.25$).

According to the information, It is also given that, X follows a normal distribution with mean =$4$ and standard deviation =$1.2$.

The sample size is n=$1.6$.

Here, X indicates the time taken to complete one review and $\stackrel{-}{x}$ represents the mean time taken to complete the $16$ reviews.

## Step 3: Explanation

Since, $X~N\left({\mu }_{x}=4,{\sigma }_{x}=1.2\right)$, the distribution of the sample mean is

$\overline{X}~N\left({\mu }_{X},\frac{{\sigma }_{X}}{\sqrt{n}}\right)\text{.Thus,}$

$\overline{X}-N\left(4,\frac{1.2}{\sqrt{16}}\right)$

$\overline{X}-N\left(4,0.3\right)$ ### Want to see more solutions like these? 