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The value of Young's modulus (GPa) was determined for cast plates consisting of certain intermetallic substrates, resulting in the following sample observations ("Strength and Modulus of a Molybdenum-Coated Ti-25Al10Nb-3U-1Mo Intermetallic," \(J\). of Materials Engr. and Performance, 1997: 46-50): \(\begin{array}{lllll}116.4 & 115.9 & 114.6 & 115.2 & 115.8\end{array}\) a. Calculate \(\bar{x}\) and the deviations from the mean. b. Use the deviations calculated in part (a) to obtain the sample variance and the sample standard deviation. c. Calculate \(s^{2}\) by using the computational formula for the numerator \(S_{x x}\) d. Subtract 100 from each observation to obtain a sample of transformed values. Now calculate the sample variance of these transformed values, and compare it to \(s^{2}\) for the original data.

Step by step solution

01

Calculate the Mean

To find the mean \(\bar{x}\), sum all observations and divide by the number of observations.\[\bar{x} = \frac{116.4 + 115.9 + 114.6 + 115.2 + 115.8}{5} = 115.58\]

02

Calculate Deviations from the Mean

Subtract the mean from each observation to find the deviations. Deviations are: \(116.4 - 115.58 = 0.82\), \(115.9 - 115.58 = 0.32\), \(114.6 - 115.58 = -0.98\), \(115.2 - 115.58 = -0.38\), \(115.8 - 115.58 = 0.22\).

03

Calculate Sample Variance and Standard Deviation

The sample variance \(s^2\) is the average of the squared deviations from the mean, divided by \(n - 1\).\[s^2 = \frac{(0.82)^2 + (0.32)^2 + (-0.98)^2 + (-0.38)^2 + (0.22)^2}{4} = 0.705\]The sample standard deviation \(s\) is the square root of the variance.\[s = \sqrt{0.705} = 0.840\]

04

Use the Computational Formula for Variance

The computational formula for the numerator \(S_{xx}\) is \(S_{xx} = \sum x_i^2 - \frac{(\sum x_i)^2}{n}\).\[S_{xx} = (116.4^2 + 115.9^2 + 114.6^2 + 115.2^2 + 115.8^2) - \frac{(577.9)^2}{5}\]Calculating gives \(S_{xx} = 2.82\) and dividing it by \(n - 1 = 4\) confirms \(s^2 = 0.705\).

05

Transform the Data and Calculate Transformed Variance

Transform the values by subtracting 100 from each: \[16.4, 15.9, 14.6, 15.2, 15.8\]Calculate the mean of the transformed data: \[\bar{x}_{\text{transformed}} = \frac{16.4 + 15.9 + 14.6 + 15.2 + 15.8}{5} = 15.58\]Find deviations \(0.82, 0.32, -0.98, -0.38, 0.22\), and calculate variance as previously: \[s^2_{\text{transformed}} = 0.705\]Since variance is independent of the shift, \(s^2\) remains the same.

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Key Concepts

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

Sample Variance

Sample variance is a way to quantify the spread of data points around the mean. It helps in understanding how much the data points deviate from the average value. To calculate the sample variance, follow these simple steps:

• First, determine the mean of your data set, which is the average of all the values.
• Next, subtract the mean from each data point to find the deviations.
• Square each of these deviations to ensure all values are positive.
• Finally, sum up all the squared deviations and divide by the number of data points minus one (n-1). This step accounts for the fact that you're working with a sample, not an entire population.

By using the formula, \[s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}\]you can easily compute the sample variance, which is crucial for many statistical analyses.

Standard Deviation

Standard deviation is derived from the sample variance and represents the typical amount by which individual data points differ from the mean. It provides a valuable insight into the variability or consistency of a data set.
To calculate the standard deviation, simply take the square root of the sample variance. This is because variance is calculated in squared units, and taking the square root converts it back to the original unit, making it easier to understand and interpret. The formula for standard deviation is:

• Calculate the sample variance (\(s^2\)).
• Use the formula \(s = \sqrt{s^2}\) to find the standard deviation.

Understanding the standard deviation helps in assessing risk and variability in different fields such as finance, science, and research. It's a key metric for determining how spread out the values in a data set are around the mean.

Mean Calculation

Mean, also known as the average, is one of the most fundamental concepts in statistics. It provides a central value for a given data set, offering a summary of the data's central tendency. Mean calculation is simple but vital for further statistical analyses.
Here’s how to calculate the mean:

• Add all the values or data points together.
• Count the number of data points in the set.
• Divide the total sum by the number of data points.

The formula is:\[\bar{x} = \frac{\sum x_i}{n}\]
The mean is used as the foundation for calculating both variance and standard deviation, making it an indispensable part of statistical analysis. Knowing the mean helps to get a general idea about the data set's overall performance and is the starting point for more intricate data analyses.