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Chapter 3: Problem 25

What are the big corporations doing with their wealth? One way to answer this question is to examine profits as percentage of assets. A random sample of 50 Fortune 500 companies gave the following information (Source: Based on information from Fortune 500, Vol. 135, No. 8). $$\begin{array}{l|ccccc} \hline \begin{array}{l} \text { Profit as percentage } \\ \text { of assets } \end{array} & 8.6-12.5 & 12.6-16.5 & 16.6-20.5 & 20.6-24.5 & 24.6-28.5 \\ \hline \begin{array}{l} \text { Number of } \\ \text { companies } \end{array} & 15 & 20 & 5 & 7 & 3 \\ \hline \end{array}$$ Estimate the sample mean, sample variance, and sample standard deviation for profit as percentage of assets.

Short Answer

Expert verified
Sample mean: 15.59, sample variance: 23.39, sample standard deviation: 4.84.

Step by step solution

01

Calculation of Midpoints

Identify the midpoints for each class interval of profit as percentage of assets. The midpoints are:- For 8.6-12.5, midpoint = \((8.6 + 12.5) / 2 = 10.55\)- For 12.6-16.5, midpoint = \((12.6 + 16.5) / 2 = 14.55\)- For 16.6-20.5, midpoint = \((16.6 + 20.5) / 2 = 18.55\)- For 20.6-24.5, midpoint = \((20.6 + 24.5) / 2 = 22.55\)- For 24.6-28.5, midpoint = \((24.6 + 28.5) / 2 = 26.55\)
02

Calculate Sample Mean

The sample mean \( \bar{x} \) is calculated using the formula:\[ \bar{x} = \frac{\sum(f_i x_i)}{n} \]where \( f_i \) is the frequency of each class, and \( x_i \) is the midpoint. Compute:\[ \bar{x} = \frac{(15 \times 10.55) + (20 \times 14.55) + (5 \times 18.55) + (7 \times 22.55) + (3 \times 26.55)}{50} \]\[ \bar{x} = \frac{(158.25 + 291 + 92.75 + 157.85 + 79.65)}{50} \]\[ \bar{x} = \frac{779.5}{50} = 15.59 \]
03

Calculate Squared Deviations

Find the squared deviations for each midpoint from the sample mean:- \((10.55 - 15.59)^2 = 25.34\)- \((14.55 - 15.59)^2 = 1.08\)- \((18.55 - 15.59)^2 = 8.83\)- \((22.55 - 15.59)^2 = 48.53\)- \((26.55 - 15.59)^2 = 120.08\)
04

Calculate Weighted Sum of Squared Deviations

Multiply each squared deviation by the class frequency and sum them up:\[ \sum(f_i(x_i - \bar{x})^2) = 15 \times 25.34 + 20 \times 1.08 + 5 \times 8.83 + 7 \times 48.53 + 3 \times 120.08 \]\[ = 380.1 + 21.6 + 44.15 + 339.71 + 360.24 = 1145.8 \]
05

Calculate Sample Variance and Standard Deviation

The sample variance \( s^2 \) is given by:\[ s^2 = \frac{\sum(f_i(x_i - \bar{x})^2)}{n-1} = \frac{1145.8}{49} \approx 23.39 \]The sample standard deviation \( s \) is the square root of the sample variance:\[ s = \sqrt{23.39} \approx 4.84 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean
The sample mean is a core concept in descriptive statistics, often used to find the average of a set of data points. In this exercise, the mean represents the average profit as a percentage of assets for the sampled Fortune 500 companies.

To calculate the sample mean, each class interval of profit is replaced by its midpoint. This midpoint simplifies calculations by offering a single value for each class.

The formula for the sample mean \( \bar{x} \) is:\[ \bar{x} = \frac{\sum(f_i x_i)}{n} \]where:
  • \( f_i \) is the frequency of the class (number of companies).
  • \( x_i \) is the midpoint of each class interval.
  • \( n \) is the total number of observations (50 companies in this case).
Here, the sample mean is computed as 15.59, meaning that, on average, companies achieve about 15.59% profit as a percentage of assets. Understanding the sample mean provides insights into the central tendency of the dataset, allowing us to summarize a large amount of data into a single, representative figure.
Sample Variance
Sample variance helps measure the spread of the data points around the sample mean. Higher variance means data points are spread out widely, while lower variance indicates they are closer to the mean.

Variance is calculated by first determining the squared deviations of each midpoint from the sample mean. Specifically, for each class interval:
  • Subtract the sample mean from the midpoint.
  • Square the result to eliminate negative values.
These squared deviations are then multiplied by the class frequency to find the weighted squared deviations.

The formula for sample variance \( s^2 \) is:\[ s^2 = \frac{\sum(f_i(x_i - \bar{x})^2)}{n-1} \]where \( n-1 \) is used instead of \( n \) to provide an unbiased estimate of the population variance. In this exercise, the sample variance is approximately 23.39, indicating the level of dispersion in the profit percentages of these companies.
Standard Deviation
The standard deviation is a measure closely related to variance, yet it is much more intuitive to interpret due to having the same units as the data.

Simply put, standard deviation provides a gauge of the average distance of data points from the sample mean. Given the variance calculated previously, the standard deviation \( s \) is the square root of the sample variance:\[ s = \sqrt{23.39} \approx 4.84 \]This tells us that most profit percentages cluster within 4.84 units of the sample mean. A smaller standard deviation would imply that profit percentages are more tightly grouped around the mean, while a larger standard deviation indicates a wider spread.

It assists in understanding the variability of the data and contextualizes how much the actual profits might differ from the average company in the sample.

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Most popular questions from this chapter

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