Q.31 - Elementary Statistics

Question

31. Mortgage Industry Employees. In an issue of National Mortgage News, a special report was published on publicly traded mortgage industry companies. A sample of $25$ mortgage industry companies had the following numbers of employees.

a. Obtain a normal probability plot of the data.
b. Use part (a) to identify any outliers.
c. Use part (a) to assess the normality of the variable under consideration.

Step-by-Step Solution

Answer

(a) The plot of the data for a normal probability as:

(b) No outliers are found.

(c) The observations will not satisfy the normality.

Part (a) Step 1: Given information

A sample of $25$ mortgage industry companies had the following numbers of employees.

260	20,800	1,801	2,073	3,596
3,223	2,128	1,796	17,540	15
29,272	6,929	2,468	7,000	6,600
2,458	3,216	209	726	9,200
650	4,800	19,400	24,886	3,082

Part (a) Step 2: Explanation

The random variable $X$'s $z-$ score is calculated as follows:
$z=\frac{x-\mu }{\sigma }$ where, $\mu$ indicates the mean and $\sigma$ indicates the standard deviation.
The mean is $\mu =6965.12$
The variance is $\sigma =8415.98759$
The $z-$ scores for the given values are listed in the table below:

Number of Employee	z-score	Number of Employee	z-score
260	-0.7967	209	-0.8028
3223	-0.4446	19400	1.47753
29272	2.65054	2073	-0.5813
2458	-0.5355	17540	1.25652
650	-0.7504	7000	0.00414
20800	1.64388	726	-0.7413
2128	-0.5748	24886	2.12939
6929	-0.0043	3596	-0.4003
3216	-0.4455	15	-0.8258
4800	-0.2573	6600	-0.0434
1801	-0.6136	9200	0.26555
1796	-0.6142	3082	-0.4614
2468	-0.5344

Part (a) Step 3: Explanation

The number of employees is plotted on the horizontal axis, while the normal scores are plotted on the vertical axis.

Part (b) Step 1: Given information

To identify the outliers by using part (a).

Part (b) Step 2: Explanation

An outlier is a data point that deviates from the plot's overall pattern.
There isn't a single observation that deviates from the plot's overall pattern.
As a result, there are no outliers.

Part (c) Step 1: Given information

To assess the normality of the variable under consideration by using part (a).

Part (c) Step 2: Explanation

The values of the sample are normally distributed if the normal probability plot is generally linear.
The plot does not follow a linear pattern.
As a result, the observations will fail to satisfy the normal requirement.