Question

Complete the distributions.

a.$X~$ _____(_____,_____)

b. role="math" localid="1648618952256" $\overline{X}~$_____(_____,_____)

Short answer

(a)$X~N(4,1.2)$

(b)$\overline{x}~N\left(4,0.3\right)$

Step-by-step solution

Given information (part a)

Given in the question that, 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 ${\rm X}$ be the random variable representing the time it takes her to complete one review. Assume ${\rm X}$ is normally distributed. Let role="math" localid="1648619084184" $\overline{x}$be the random variable representing the mean time to complete the re$16$views. Assume that the$16$reviews represent a random set of reviews.

Explanation(part a)

According to the provided details, the time for each review $\left(X\right)$is normally distributed, the meantime for each review is $4$hours and the standard deviation is $1.2$hours and the sample size is $16$employees.

The normal distribution with mean $\left(\mu \right)$and standard deviation $\left(\sigma \right)$is given as:

$X~N(\mu ,\sigma )$

So, the distribution time for each review $\left(X\right)$is given as below:

$X~N(4,1.2)$

Final answer(part a)

$X~N(4,1.2)$

Given information(part b)

Given in the question that, 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${\rm X}$be the random variable representing the time it takes her to complete one review. Assume ${\rm X}$is normally distributed. Let $\overline{x}$be the random variable representing the mean time to complete the $16$reviews. Assume that the $16$reviews represent a random set of reviews.

Explanation(part b)

According to the provided details, the time for each review$\left(X\right)$is normally distributed, the meantime for each review is $4$hours and the standard deviation is $1.2$hours and the sample size is $16$employees.

The distribution for the sample mean $\left(\overline{X}\right)$is given as below:

role="math" localid="1648622435181" $\overline{X}~N\left(\mu x,\frac{\sigma x}{\sqrt{n}}\right)$

role="math" localid="1648622812034" $\overline{X}~N\left(4,\frac{1.2}{\sqrt{16}}\right)$

$\overline{x}~N\left(4,0.3\right)$

Final answer(part b)

$\overline{x}~N\left(4,0.3\right)$