Q. 6.140 - Elementary Statistics

Question

Desert Samaritan Hospital in Mesa, Arizona, keeps records of emergency room traffic. Those records reveal that the times between arriving patients have a special type of reverse-J-shaped distribution called an exponential distribution. The records also show that the mean time between arriving patients is $8.7$8 minutes.

a. Use the technology of your choice to simulate four random samples of $75$ interarrival times each.

b. Obtain a normal probability plot of each sample in part (a).

c. Are the normal probability plots in part (b) what you expected? Explain your answer.

Step-by-Step Solution

Answer

a. The simulate four random samples of patient's samples are $x=-\log (1-rand(75,4\left)\right)\times 8.7$

b. A normal probability plot of each sample the plot will be,

c.  As we can see in the figure the population distribution is normally distributed. The plot is not regular linear, the variables are not roughly normally distributed. 

Part (a) Step 1: Given Information

To explain simulate the random patients which has $75$ interarrival time for each with$8.7$ mean time. The number of samples and mean of arrival. 

$n=75$

Mean $=8.7$.

Part (a) Step 2: Explanation

Let's take $n=75$

Then sing MATLAB create a random matrix which has the mean $8.7$

We will use the function

$x=F^{-1}(p\mid \mu )$

$=-\mu \ln (1-p)$

Here $\mu$ is the mean of arrival time and $p$ is the random sample.

$p=$rand$(75,4)$

$\mu =8.7$

Put all the values into the above equation and get the random 4 samples such as

$x=-\ln (1-rand(75,4\left)\right)\times 8.7$

After solving the equation, we will get the answer.

Part (a) Step 3: Explanation

Program:

Query: 

We started by determining the quantity of samples.

Then make a matrix with an average arrival time of $8.7$. 

We shall arrive at a solution after simplifying.

Part (b) Step 1: Given Information

To determine the create a normal probability plot for the random sample from part (a).

The number of samples and mean of arrival is given.

$n=75$

Mean$=8.7$.

Part (b) Step 2: Explanation

Then sing MATLAB create a random matrix which has the mean $8.7$

We will use the function

$x=F^{-1}(p\mid \mu )$

$=-\mu \ln (1-p)$

Here $\mu$ is the mean of arrival time and $p$ is the random sample.

$p=rand(75,4)$

$\mu =8.7$

Put all the values into the above equation and get the random $4$ samples such as

$x=-\ln (1-rand(75,4\left)\right)\times 8.7$

After solving the equation, we will get the answer.

Part (b) Step 3: Explanation

Program:

Query:

We began by calculating the number of samples required.

Make a matrix with an $8.7$ average arrival time.

After simplifying, we'll arrive at a solution.

Draw a graph of the samples' normal probability distribution.

Part (c) Step 1: Given Information

Explain your answer what would you expect from normal probability plot from part (b).

Part (c) Step 2: Explanation

The figure the population distribution is normally distributed.

If the probability plot is regular linear, the variables are roughly normally distributed; if the plot is not regular linear, the variables are not roughly normally distributed.