91Ó°ÊÓ

Kevlar epoxy is a material used on the NASA Space Shuttle. Strands of this epoxy were tested at the \(90 \%\) breaking strength. The following data represent time to failure (in hours) for a random sample of 50 epoxy strands (Reference: \(\mathrm{R} . \mathrm{E} .\) Barlow, University of California, Berkeley). Let \(x\) be a random variable representing time to failure (in hours) at \(90 \%\) breaking strength. Note: These data are also available for download online in HM StatSPACETM. \(\begin{array}{llllllllll}0.54 & 1.80 & 1.52 & 2.05 & 1.03 & 1.18 & 0.80 & 1.33 & 1.29 & 1.11 \\ 3.34 & 1.54 & 0.08 & 0.12 & 0.60 & 0.72 & 0.92 & 1.05 & 1.43 & 3.03 \\ 1.81 & 2.17 & 0.63 & 0.56 & 0.03 & 0.09 & 0.18 & 0.34 & 1.51 & 1.45 \\ 1.52 & 0.19 & 1.55 & 0.02 & 0.07 & 0.65 & 0.40 & 0.24 & 1.51 & 1.45 \\ 1.60 & 1.80 & 4.69 & 0.08 & 7.89 & 1.58 & 1.64 & 0.03 & 0.23 & 0.72\end{array}\) (a) Find the range. (b) Use a calculator to verify that \(\Sigma x=62.11\) and \(\Sigma x^{2}=164.23\). (c) Use the results of part (b) to compute the sample mean, variance, and standard deviation for the time to failure. (d) Use the results of part (c) to compute the coefficient of variation. What does this number say about time to failure? Why does a small CV indicate more consistent data, whereas a larger CV indicates less consistent data? Explain.

Step by step solution

01

Range Calculation

The range is found by subtracting the smallest value from the largest value in the dataset. In the given data, the smallest value is 0.02 and the largest value is 7.89. Therefore, the range is \(7.89 - 0.02 = 7.87\).

02

Verification of Sigma Values

To verify the given values, you sum all the data points to find \(\Sigma x\), and then square each data point, sum them, and verify \(\Sigma x^2\). The given values \(\Sigma x = 62.11\) and \(\Sigma x^2 = 164.23\) are to be verified by calculations which should match the totals obtained from the dataset.

03

Compute Sample Mean

The sample mean \(\bar{x}\) is calculated using \(\bar{x} = \frac{\Sigma x}{n}\), where \(n = 50\). Thus, \(\bar{x} = \frac{62.11}{50} = 1.2422\).

04

Calculate Variance

The sample variance \(s^2\) is calculated by \(s^2 = \frac{\Sigma x^2 - \frac{(\Sigma x)^2}{n}}{n-1}\). Substituting in the given values, \(s^2 = \frac{164.23 - \frac{(62.11)^2}{50}}{49} = 1.6854\).

05

Calculate Standard Deviation

The standard deviation \(s\) is the square root of the variance. Therefore, \(s = \sqrt{1.6854} \approx 1.2982\).

06

Coefficient of Variation Calculation

The coefficient of variation (CV) is calculated as \( \text{CV} = \frac{s}{\bar{x}} \times 100\% \), which provides a measure of relative variability. Using the sample mean \(\bar{x} = 1.2422\) and \(s = 1.2982\), \( \text{CV} = \frac{1.2982}{1.2422} \times 100\% \approx 104.51\%\).

07

Interpretation of Coefficient of Variation

A high CV (104.51%) indicates that there is a high level of disparity or variance relative to the mean time to failure. A smaller CV would suggest the dataset has more precise and consistent measurements relative to its means, whereas a larger CV suggests greater variability and less consistency.

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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 fundamental measure in statistics that gives the central value of a dataset. It is essentially the average of all the observations. To find the sample mean, simply add up all the values in your dataset and then divide by the number of observations.

In our example, we have 50 strands of Kevlar epoxy, each with a recorded time to failure. By summing all these times, we get a total of 62.11 hours. Dividing this by the number of data points, 50, gives us a sample mean \(\bar{x} = \frac{62.11}{50} = 1.2422\) hours.

Understanding the sample mean is crucial because it gives you a quick sense of where the bulk of your data points are located. It provides a baseline value to compare individual observations against.

Variance

Variance is a statistical measure that describes the spread or dispersion of a dataset. It tells us how much the individual data points differ from the mean. A higher variance indicates that the data points are more spread out from the mean, while a lower variance indicates they are closer.

The formula for sample variance is\[s^2 = \frac{\Sigma x^2 - \frac{(\Sigma x)^2}{n}}{n-1}\]
where:

• \(\Sigma x^2\) is the sum of the squares of the observations,
• \(\Sigma x\) is the sum of the observations,
• \(n\) is the number of observations.

For our dataset, substituting the values, we get: \[s^2 = \frac{164.23 - \frac{(62.11)^2}{50}}{49} = 1.6854\]
Understanding variance helps in assessing the consistency of the data around the mean.

Standard Deviation

The standard deviation is a widely used measure that represents the average distance of each data point from the mean. It is simply the square root of the variance, making it more interpretable since it is in the same unit as the data itself.

For our dataset, the standard deviation is calculated as follows:\[s = \sqrt{1.6854} \approx 1.2982\]
This value helps to quickly grasp how data points spread around the mean. If the standard deviation is small, most data points cluster close to the mean. Conversely, a large standard deviation suggests a wide spread around the mean.

Standard deviation is a fundamental tool in statistics because it offers insights into data consistency and variability at a glance.

Coefficient of Variation

The coefficient of variation (CV) is a standardized measure of dispersion that expresses the standard deviation as a percentage of the mean. This allows for comparison of variability between datasets with different units or scales.

To calculate the CV, use the formula:\[\text{CV} = \frac{s}{\bar{x}} \times 100\%\]
where \(s\) is the standard deviation and \(\bar{x}\) is the mean.
In our case, the CV is:\[\text{CV} = \frac{1.2982}{1.2422} \times 100\% \approx 104.51\%\]
This high CV total indicates a substantial level of variability in relation to the mean. In practice, lower CV values suggest more consistent data, while higher CV values highlight greater variation. Understanding CV helps researchers and engineers assess the reliability and quality of data in different contexts or units.