Question

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 $\stackrel{-}{x}$be the random variable representing the meantime to complete the $16$reviews. Assume that the $16$reviews represent a random set of reviews.

Find the probability that one review will take Yoonie from $3.5$to $4.25$hours. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability

b. P(________ <x< ________) = _______

Short answer

From the information above

a) The probability that one review will take Yoonie from $3.5$to $4.25$hours is $0.2459$. and the shaded region corresponding to the probability is:

b)$p(3.5<x<4.25)=0.2459$

Step-by-step solution

Given Information (part a)

It is provided that Yoonie reviews $16$of the employees with an average time of $4$hours and a standard deviation of $1.2$hours. We have to find the probability that Yoonie will take $3.5$to $4.25$hours for one review and graph it.

Step 2: Explanation(part a)

According to the information, we have to find the probability that one review will take Yoonie from 3.5 to 4. 25

The Computation of the probability is given as below:

$p(3.5<x<4.25)=p(Z<\frac{4.5-\mu }{\sigma })-p(Z<\frac{3.5-\mu }{\sigma })$

$=p(Z<\frac{4.25-4}{1.2})-p(Z<\frac{3.5-4}{1.2})$

$=p(Z<0.21)-p(Z<-0.42)$

Therefore,

localid="1648621489396" $p(z<0.21)$and

$p(Z<-0.42)\text{in Excel is:}$

Step 3: Shaded region representation (part a)

Therefore,

$p(3.5<x<4.25)=p(Z<0.21)-p(Z<-1.42)$

$=0.5832-0.3372$

=$0.2459$

The shaded region corresponding to the probability is as shown below:

Given Information (part b)

It is provided that Yoonie reviews $16$of the employees with an average time of $4$hours and a standard deviation of$1.2$ hours

Explanation (part b)

The probability is

$p(3.5<x<4.25)=p(Z<0.21)-p(Z<-1.42)$

$=0.5832-0.3372$

Simplify,

localid="1648621737422" $=0.2459$