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Chapter 3: Problem 42

Do the following: a. Compute the sample variance. b. Determine the sample standard deviation. The sample of eight companies in the aerospace industry, referred to in Exercise \(30,\) was surveyed as to their return on investment last year. The results are \(10.6,12.6,14.8,18.2,12.0,14.8,12.2,\) and 15.6.

Short Answer

Expert verified
Sample variance is 6.03. Sample standard deviation is 2.46.

Step by step solution

01

Calculating the Mean

First, compute the mean (average) of the sample data. Add all the data points together and divide by the number of data points. The data points are: 10.6, 12.6, 14.8, 18.2, 12.0, 14.8, 12.2, and 15.6. Mean = \( \frac{10.6 + 12.6 + 14.8 + 18.2 + 12.0 + 14.8 + 12.2 + 15.6}{8} = \frac{111.8}{8} = 13.975 \)
02

Calculating the Deviation from the Mean

Subtract the mean from each data point to find the deviation of each point:- \(10.6 - 13.975 = -3.375\)- \(12.6 - 13.975 = -1.375\)- \(14.8 - 13.975 = 0.825\)- \(18.2 - 13.975 = 4.225\)- \(12.0 - 13.975 = -1.975\)- \(14.8 - 13.975 = 0.825\)- \(12.2 - 13.975 = -1.775\)- \(15.6 - 13.975 = 1.625\)
03

Squaring the Deviations

Square each deviation to eliminate negative signs:- \((-3.375)^2 = 11.390625\)- \((-1.375)^2 = 1.890625\)- \((0.825)^2 = 0.680625\)- \((4.225)^2 = 17.850625\)- \((-1.975)^2 = 3.900625\)- \((0.825)^2 = 0.680625\)- \((-1.775)^2 = 3.150625\)- \((1.625)^2 = 2.640625\)
04

Calculating Sample Variance

Calculate the sample variance by adding all the squared deviations together and then dividing by \(n-1\) (where \(n\) is the number of data points) to correct for sample bias.Sample Variance = \(\frac{11.390625 + 1.890625 + 0.680625 + 17.850625 + 3.900625 + 0.680625 + 3.150625 + 2.640625}{7}\)Sample Variance = \(\frac{42.185}{7} = 6.026429\)
05

Calculating Sample Standard Deviation

The sample standard deviation is the square root of the sample variance:Sample Standard Deviation = \(\sqrt{6.026429} \approx 2.455\)

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Key Concepts

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

Sample Standard Deviation
Sample Standard Deviation helps us understand how much the data varies from the average. It's calculated from the sample variance, which indicates the spread of data points.
The formula for sample standard deviation is \[S = \sqrt{\text{Sample Variance}}\] In this case, after finding the sample variance as outlined, \(6.026429\), the sample standard deviation is calculated by taking its square root.
  • This results in approximately \(2.455\), which tells us the average deviation from the mean.
  • The smaller the standard deviation, the closer the data points are to the mean, indicating more consistency in the dataset.

Understanding this helps in assessing risk and variability in fields like finance, where consistency can be crucial.
Mean Calculation
The Mean Calculation is crucial as it provides a single value that represents the center of the dataset. To find the mean, sum all the data points.
Divide this sum by the number of observations. For our sample:
  • Addition: \(10.6 + 12.6 + 14.8 + 18.2 + 12.0 + 14.8 + 12.2 + 15.6 = 111.8\)
  • Division by the number of data points: \(\frac{111.8}{8} = 13.975\)

This mean \(13.975\) functions as a baseline for assessing how data points deviate. It is essential in understanding trends, such as the average financial return or business performance comparisons.
Squared Deviations
Squared Deviations are used to eliminate negative values when calculating variability.
Factors to consider:
  • Compute deviations by subtracting the mean from each data point.
  • Square each result to ensure all numbers are positive.
  • This squaring helps to amplify larger deviations, which can be significant in data analysis.
For our example data, these squared deviations look like this: \[ (-3.375)^2, (-1.375)^2, (0.825)^2,...\] This method helps bring to light how spread out or clustered the data points are relative to the mean.
Return on Investment Analysis
Return on Investment (ROI) is a measure that evaluates the profitability of an investment.
In a business context, ROI is crucial for understanding how successful an investment has performed. It is calculated as:\[\text{ROI} = \frac{\text{Net Profit}}{\text{Total Investment}} \times 100\] In the context of our exercise:
  • Each company's ROI is considered a data point.
  • Analyzing how these ROIs compare to the average can identify industry trends.

Consistent ROIs across companies might suggest steady economic conditions. However, larger variability could indicate risk or areas for improvement. Understanding the spread and central tendency of ROIs helps businesses plan and optimize future investments.

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Most popular questions from this chapter

Big Orange Trucking is designing an information system for use in 鈥渋n-cab" communications. It must summarize data from eight sites throughout a region to describe typical conditions. Compute an appropriate measure of central location for the variables wind direction, temperature, and pavement. $$ \begin{array}{|llcl|} \hline \text { City } & \text { Wind Direction } & \text { Temperature } & \text { Pavement } \\ \hline \text { Anniston, AL } & \text { West } & 89 & \text { Dry } \\ \text { Atlanta, GA } & \text { Northwest } & 86 & \text { Wet } \\ \text { Augusta, GA } & \text { Southwest } & 92 & \text { Wet } \\ \text { Birmingham, AL } & \text { South } & 91 & \text { Dry } \\ \text { Jackson, MS } & \text { Southwest } & 92 & \text { Dry } \\ \text { Meridian, MS } & \text { South } & 92 & \text { Trace } \\ \text { Monroe, LA } & \text { Southwest } & 93 & \text { Wet } \\ \text { Tuscaloosa, AL } & \text { Southwest } & 93 & \text { Trace } \end{array} $$

Consider these six values a population: \(13,3,8,10,8,\) and 6 a. Determine the mean of the population. b. Determine the variance.

In June, an investor purchased 300 shares of Oracle (an information technology company) stock at \(\$ 41\) per share. In August, she purchased an additional 400 shares at \(\$ 39\) per share. In November, she purchased an additional 400 shares at \(\$ 45\) per share. What is the weighted mean price per share?

The annual incomes of the five vice presidents of TMV Industries are \(\$ 125,000 ;\) \(\$ 128,000 ; \$ 122,000 ; \$ 133,000 ;\) 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.

Calculate the (a) range, (b) arithmetic mean, and (c) variance, and (d) interpret the statistics. All eight companies in the aerospace industry were surveyed as to their return on investment last year. The results are: \(10.6 \%, 12.6 \%, 14.8 \%, 18.2 \%, 12.0 \%\) \(14.8 \%, 12.2 \%,\) and \(15.6 \%\).
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