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Chapter 6: Problem 10

An article in the Journal of Structural Engineering (Vol. 115,1989 ) describes an experiment to test the yield strength of circular tubes with caps welded to the ends. The first yields (in \(\mathrm{kN}\) ) are 96,96,102,102,102,104,104,108,126,126,128,128 \(140,156,160,160,164,\) and \(170 .\) Calculate the sample mean and sample standard deviation. Construct a dot diagram of the data.

Short Answer

Expert verified
Sample mean is approximately 120.78; sample standard deviation is approximately 13.21.

Step by step solution

01

Calculate Sample Mean

To find the sample mean, sum all the data values, and then divide by the number of data points. The data values are: 96, 96, 102, 102, 102, 104, 104, 108, 126, 126, 128, 128, 140, 156, 160, 160, 164, 170. Add them: 96 + 96 + 102 + 102 + 102 + 104 + 104 + 108 + 126 + 126 + 128 + 128 + 140 + 156 + 160 + 160 + 164 + 170 = 2174. Count the number of data points: 18. So, the sample mean is \( \frac{2174}{18} \approx 120.78 \).
02

Calculate Sample Standard Deviation

To calculate the sample standard deviation, first find the mean (\( \mu = 120.78 \)). Then find the squared difference from the mean for each data point. Sum these squared differences and divide by \(n-1\) (where \(n\) is the number of data points) to find the variance, and then take the square root of the variance to find the standard deviation. Calculate the squared differences, sum them, divide by 17 (\(18-1\)) to get the variance: \(2957.56/17 \approx 174.56\). Then, the standard deviation is \(\sqrt{174.56} \approx 13.21\).
03

Construct Dot Diagram

To construct a dot diagram, place a dot for each data point along a number line at its appropriate value. For multiple occurrences of the same value, stack the dots vertically. Plot these for the data: 96, 96, 102, 102, 102, 104, 104, 108, 126, 126, 128, 128, 140, 156, 160, 160, 164, and 170. Place two dots above 96, three above 102, two above 104, one above 108, two each above 126 and 128, one above 140, one above 156, two above 160, one above 164, and one above 170.

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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 concept in statistics. It provides a simple way to summarize a set of data points with a single value that represents the "center" of the data. To calculate the sample mean, you need to add up all the individual data values and then divide the total by how many data points there are.

Here's how you can calculate it:
  • First, add all the values together. For example, in the given data set, you would add: 96+96+102+102+102+104+104+108+126+126+128+128+140+156+160+160+164+170 = 2174.
  • Next, count the total number of data points, which in this case is 18.
  • Finally, divide the sum by the number of data points to get the sample mean: \(\frac{2174}{18} \approx 120.78\).
The sample mean gives us a useful "average" value, helping summarize the general size of the yield strengths from the experiment.
Sample Standard Deviation
The sample standard deviation is another essential statistical measure. It shows how much the data values are spread out from the mean. A higher standard deviation means the data is more spread out, while a lower standard deviation means it's more tightly packed around the mean.

To calculate the sample standard deviation, follow these steps:
  • First, find the mean (in this case, 120.78).
  • Next, for each data point, calculate its deviation from the mean, square that deviation, and then add all those squared deviations together.
  • Divide this sum by \(n-1\), where \(n\) is the number of data points, which gives you the variance. For our example, it would be \(\frac{2957.56}{17} \approx 174.56\).
  • Finally, take the square root of the variance to get the sample standard deviation, which is \(\sqrt{174.56} \approx 13.21\).
Understanding the variation in data is crucial because it helps us gauge whether the average we calculated is truly representative of the data set.
Dot Diagram
A dot diagram, also known as a dot plot, is a simple yet powerful tool for visualizing the distribution of data. It provides a clear way to see the frequency of each value and the spread of the data over the range of possible values.

To construct a dot diagram:
  • Draw a number line that includes the range of your data. In this case, from 96 to 170.
  • Place a dot above the number line for each data point at its corresponding value. For repeated values, stack the dots vertically.
  • For example, two dots are placed above 96, three dots above 102, and so on.
This graphical representation helps you quickly identify clusters, gaps, and any possible outliers in your data. It's an excellent visual aid for understanding the overall shape and distribution of the data set.

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Most popular questions from this chapter

An article in the Journal of Aircraft (1988) described the computation of drag coefficients for the NASA 0012 airfoil. Different computational algorithms were used at \(M_{\infty}=0.7\) with the following results (drag coefficients are in units of drag counts; that is, one count is equivalent to a drag coefficient of 0.0001)\(: 79,100,74,83,81,85,82,80,\) and \(84 .\) Compute the sample mean, sample variance, and sample standard deviation, and construct a dot diagram.

An experiment to investigate the survival time in hours of an electronic component consists of placing the parts in a test cell and running them for 100 hours under elevated temperature conditions. (This is called an "accelerated" life test.) Eight components were tested with the following resulting failure times: \(75,63,100^{+}, 36,51,45,80,90\) The observation \(100^{+}\) indicates that the unit still functioned at 100 hours. Is there any meaningful measure of location that can be calculated for these data? What is its numerical value?

An article in the Journal of Physiology ["Response of Rat Muscle to Acute Resistance Exercise Defined by Transcriptional and Translational Profiling" (2002, Vol. 545, pp. \(27-41\) ) ] studied gene expression as a function of resistance exercise. Expression data (measures of gene activity) from one gene are shown in the following table. One group of rats was exercised for six hours while the other received no exercise. Compute the sample mean and standard deviation of the exercise and no-exercise groups separately. Construct a dot diagram for the exercise and no-exercise groups separately. Comment on any differences for the groups.

The January 1990 issue of Arizona Trend contains a supplement describing the 12 "best" golf courses in the state. The yardages (lengths) of these courses are as follows: 6981 , 7099,6930,6992,7518,7100,6935,7518,7013,6800,7041 and \(6890 .\) Calculate the sample mean and sample standard deviation. Construct a dot diagram of the data.

The following data are the viscosity measurements for a chemical product observed hourly (read down, then left to right). Construct and interpret either a digidot plot or a separate stem-and-leaf and time series plot of these data. Specifications on product viscosity are at \(48 \pm 2 .\) What conclusions can you make about process performance? $$\begin{array}{llllll}47.9 & 48.6 & 48.0 & 48.1 & 43.0 & 43.2 \\ 47.9 & 48.8 & 47.5 & 48.0 & 42.9 & 43.6 \\\48.6 & 48.1 & 48.6 & 48.3 & 43.6 & 43.2 \\ 48.0 & 48.3 & 48.0 & 43.2 & 43.3 & 43.5 \\\48.4 & 47.2 & 47.9 & 43.0 & 43.0 & 43.0 \\ 48.1 & 48.9 & 48.3 & 43.5 & 42.8 & \\\48.0 & 48.6 & 48.5 & 43.1 & 43.1 &\end{array}$$
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