Problem 46 - Basic Statistics for Business and Economics with Student

Question

46\. The annual incomes of the five vice presidents of TMV Industries are: \(\$ 125,000 ; \quad \$ 128,000 ; \quad \$ 122,000 ;\) \(\$ 133,000 ;\) and \(\$ 140,000 .\) Consider this a population. a. What is the range? b. What is the arithmetic mean income? c. What is the population variance? The standard deviation? d. The annual incomes of officers of another firm similar to TMV Industries were also studied. The mean was \(\$ 129,000\) and the standard deviation \(\$ 8,612 .\) Compare the means and dispersions in the two firms.

Step-by-Step Solution

Answer

a. Range: $18,000 b. Mean: $129,600 c. Variance: $40,240,000, Std. Dev.: $6,344 d. TMV has higher mean, lower dispersion.

Step 1: Calculate the Range

The range is calculated by subtracting the smallest annual income from the largest annual income in the population. Largest income: $140,000 Smallest income: $122,000 Range = $140,000 - $122,000 = $18,000.

Step 2: Calculate the Arithmetic Mean

The arithmetic mean is calculated by summing all the incomes and dividing by the total number of incomes. Sum of incomes: $125,000 + $128,000 + $122,000 + $133,000 + $140,000 = $648,000 Number of incomes: 5 Arithmetic Mean = $648,000 / 5 = $129,600.

Step 3: Calculate the Population Variance

To find the population variance, use the formula:\[\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}\]Where \(x_i\) are the incomes, \(\mu\) is the mean, and \(N\) is the population size.1. Calculate each deviation squared: - \((125,000 - 129,600)^2 = 21,160,000\) - \((128,000 - 129,600)^2 = 2,560,000\) - \((122,000 - 129,600)^2 = 57,760,000\) - \((133,000 - 129,600)^2 = 11,560,000\) - \((140,000 - 129,600)^2 = 108,160,000\)2. Sum of squared deviations: - Total = 21,160,000 + 2,560,000 + 57,760,000 + 11,560,000 + 108,160,000 = 201,200,0003. Divide by population size (5): - Population variance = 201,200,000 / 5 = 40,240,000.

Step 4: Calculate the Standard Deviation

The standard deviation is the square root of the variance.\(\sigma = \sqrt{40,240,000} \approx 6,344\)The population standard deviation is approximately $6,344.

Step 5: Compare the Means and Dispersions

Compare the arithmetic mean and standard deviation of TMV Industries with the other firm. TMV Industries: - Mean: $129,600 - Standard deviation: $6,344 Other firm: - Mean: $129,000 - Standard deviation: $8,612 The mean of TMV Industries is slightly higher whereas the dispersion as measured by standard deviation is lower compared to the other firm.

Key Concepts

Arithmetic Mean

The Arithmetic Mean, often referred to as the average, is a measure of central tendency that gives us an idea of where the center of a set of numbers is located. It is calculated by adding together all the numbers in a dataset and then dividing that sum by the number of data points.

For example, in the given exercise, the annual incomes of the five vice presidents at TMV Industries are summed to get $648,000. To find the Arithmetic Mean, we divide this total by the number of incomes, which is 5, resulting in an Arithmetic Mean of $129,600.

This mean provides a simple overview of the income level of the vice presidents but doesn't give us information about the variability of their incomes around this central value.

Population Variance

Population Variance is a key statistical measure used to quantify the degree of spread in a set of numbers within an entire population. It helps us understand how each value in the dataset deviates on average from the mean.

To calculate the Population Variance, you first find the deviation of each data point from the mean, square those deviations to avoid negative values, and then average those squared deviations.

Using the example provided, we found the variances around the mean of $129,600 for each income, summed these variances, and then divided by the population size, which is 5. This provided a Population Variance of 40,240,000.

In essence, the Population Variance gives us a detailed view of how varied or clustered the incomes are relative to the mean.

Standard Deviation

Standard Deviation is a critical statistic giving insight into the degree of variation or spread of a set of values. It's derived from the square root of the Population Variance, making it easier to interpret since it is expressed in the same units as the original data.

In the TMV Industries example, the Population Variance was calculated as 40,240,000, resulting in a Standard Deviation of approximately $6,344 when we took its square root.

The Standard Deviation is beneficial because:

• It indicates how much the incomes differ on average from the mean.
• It allows comparison of variability between datasets of different scales.

Essentially, a lower standard deviation indicates that the data points tend to be closer to the mean, while a higher standard deviation shows more spread out data.

Range

The Range is the simplest measure of dispersion and gives a quick snapshot of the spread of a dataset. It is determined by subtracting the smallest data point from the largest one within a dataset.

For the TMV Industries example, the Range was found by taking the highest income ($140,000) and subtracting the lowest income ($122,000), leading to a Range of $18,000.

While easy to compute, the Range only considers the extreme values and does not account for how data is distributed. It is best used in conjunction with other statistical measures such as the Standard Deviation to provide a fuller picture of data variability.