Q.5 Yoonie is a personnel manager in... [FREE SOLUTION]

91影视

Introductory Statistics

Barbara Illowsky, Susan Dean

$Math Studyset 91影视 Explanations$ Math

OER 2018 Edition 路 902 pages

ISBN: 9781938168208

Chapter 7: Q.5 (page 427)

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 围 be the random variable representing the time it takes her to complete one review. Assume 围 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?

Short Answer

Expert verified

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

Step by step solution

Given Information

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

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( $3.5$ < $\bar{x}$ < $4.25$ ).

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 $\bar{x}$ represents the mean time taken to complete the $16$ reviews.

Explanation

Since, $X ~ N (渭_{x} = 4, 蟽_{x} = 1.2)$ , the distribution of the sample mean is

$\overset{炉}{X} ~ N (渭_{X}, \frac{蟽_{X}}{\sqrt{n}}) .Thus,$

$\overset{炉}{X} 鈭� N (4, \frac{1.2}{\sqrt{16}})$

$\overset{炉}{X} 鈭� N (4, 0.3)$

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91影视

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