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Chapter 1: Problem 18

Observations on burst strength \(\left(\mathrm{lb} / \mathrm{in}^{2}\right)\) were obtained both for test nozzle closure welds and for production cannister nozzle welds ("Proper Procedures Are the Key to Welding Radioactive Waste Cannisters," Welding J., Aug. 1997: \(61-67)\) \(\begin{array}{lllllll}\text { Test } & 7200 & 6100 & 7300 & 7300 & 8000 & 7400 \\ & 7300 & 7300 & 8000 & 6700 & 8300 & \\ \text { Cannister } & 5250 & 5625 & 5900 & 5900 & 5700 & 6050 \\ & 5800 & 6000 & 5875 & 6100 & 5850 & 6600\end{array}\) Construct a comparative boxplot and comment on interesting features (the cited article did not include such a picture, but the authors commented that they had looked at one).

Short Answer

Expert verified
The test welds show more variability with a symmetric distribution, while the cannister welds have less variability and a slight left skew.

Step by step solution

01

Organize the Data

List the given data for both test welds and cannister welds. You have the following data: - **Test Welds**: 7200, 6100, 7300, 7300, 8000, 7400, 7300, 7300, 8000, 6700, 8300. - **Cannister Welds**: 5250, 5625, 5900, 5900, 5700, 6050, 5800, 6000, 5875, 6100, 5850, 6600.
02

Calculate Quartiles

Calculate the first quartile (Q1), median (Q2), and third quartile (Q3) for both data sets. - **Test Welds**: - Sort data: 6100, 6700, 7200, 7300, 7300, 7300, 7300, 7400, 8000, 8000, 8300. - Calculate: Q1 = 7200, Median (Q2) = 7300, Q3 = 8000. - **Cannister Welds**: - Sort data: 5250, 5625, 5700, 5800, 5850, 5875, 5900, 5900, 6000, 6050, 6100, 6600. - Calculate: Q1 = 5700, Median (Q2) = 5900, Q3 = 6050.
03

Identify Extremes and IQR

Determine the minimum and maximum for each dataset, and compute the interquartile range (IQR). - **Test Welds**: - Minimum = 6100, Maximum = 8300, IQR = Q3 - Q1 = 8000 - 7200 = 800. - **Cannister Welds**: - Minimum = 5250, Maximum = 6600, IQR = Q3 - Q1 = 6050 - 5700 = 350.
04

Construct Boxplots

Using the calculated quartiles, minimum, and maximum values, construct a boxplot for each dataset. Draw the box from Q1 to Q3 with a line at the median. Extend whiskers from the box to the min and max values.
05

Analyze Boxplots

Observe and comment on the features of boxplots: - **Test Welds**: median is near the center of the box, indicating symmetrical data. The range and IQR are larger than the cannister welds, indicating greater variability. - **Cannister Welds**: median is slightly closer to the top, indicating possible left skewness. The smaller IQR suggests less variability compared to test welds.

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Key Concepts

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

Quartiles
Understanding quartiles is a crucial step in data analysis, especially when creating a boxplot. Quartiles divide your data set into four equal parts. These are:
  • First Quartile (Q1): Also known as the lower quartile, it separates the lowest 25% of the data from the rest. You can find it by determining the median of the first half of your data.

  • Second Quartile (Q2): This is the median of the entire data set, effectively splitting your data into two equal parts. It divides the data into halves.

  • Third Quartile (Q3): Also known as the upper quartile, it separates the lowest 75% of the data from the top 25%. It is the median of the upper half of the data.
In our exercise, for the test welds, the quartiles are calculated as Q1 = 7200, Q2 = 7300, and Q3 = 8000. For the cannister welds, they are Q1 = 5700, Q2 = 5900, and Q3 = 6050. These quartiles are essential for constructing boxplots, as they determine the boundaries of the box.
Interquartile Range (IQR)
The interquartile range (IQR) is a measure of statistical dispersion and is one way to gauge the spread of your data. It is the difference between the third and first quartiles, indicating the spread of the middle 50% of the data. This range is a useful metric as it is less affected by outliers or skewed data.To calculate the IQR:
  • Find Q1 and Q3.

  • Subtract Q1 from Q3: \( \text{IQR} = Q3 - Q1 \).
In the given exercise, the IQR for the test welds is 800, calculated as 8000 (Q3) minus 7200 (Q1). For the cannister welds, the IQR is 350, computed as 6050 (Q3) minus 5700 (Q1). These values show the variability within the middle section of the datasets and help in identifying the spread around the median.
Data Analysis
Data analysis involves interpreting and understanding data, which in turn assists in making informed decisions. One effective tool in data analysis is the boxplot, which visually represents the distribution of data based on a five-number summary: minimum, Q1, median, Q3, and maximum. When analyzing boxplots:
  • The position of the median in the box tells you about the symmetry or skewness of the data.

  • The length of the whiskers can indicate the spread and potential outliers.

  • Comparing boxplots helps in determining differences between multiple datasets.
In this context, the analysis of test welds and cannister welds boxplots reveals interesting differences. For example, the test welds show more variability with a larger range and IQR compared to cannister welds. The different positions of the medians provide insight into the symmetry or skewness of each dataset. This simple visual interpretation can greatly aid in grasping the underlying structure of the data.

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Most popular questions from this chapter

a. Let \(a\) and \(b\) be constants and let \(y_{i}=a x_{i}+b\) for \(i=1,2, \ldots, n\). What are the relationships between \(\bar{x}\) and \(\bar{y}\) and between \(s_{x}^{2}\) and \(s_{y}^{2}\) ? b. A sample of temperatures for initiating a certain chemical reaction yielded a sample average \(\left({ }^{\circ} \mathrm{C}\right)\) of \(87.3\) and a sample standard deviation of \(1.04\). What are the sample average and standard deviation measured in \({ }^{\circ} \mathrm{F}\) ? [Hint: $$ F=\frac{9}{5} C+32 \text {.] } $$

A study of the relationship between age and various visual functions (such as acuity and depth perception) reported the following observations on the area of scleral lamina \(\left(\mathrm{mm}^{2}\right)\) from human optic nerve heads ("Morphometry of Nerve Fiber Bundle Pores in the Optic Nerve Head of the Human," Experimental Eye Research, 1988: 559-568): \(\begin{array}{lllllllll}2.75 & 2.62 & 2.74 & 3.85 & 2.34 & 2.74 & 3.93 & 4.21 & 3.88 \\ 4.33 & 3.46 & 4.52 & 2.43 & 3.65 & 2.78 & 3.56 & 3.01 & \end{array}\) a. Calculate \(\sum x_{i}\) and \(\sum x_{i}^{2}\). b. Use the values calculated in part (a) to compute the sample variance \(s^{2}\) and then the sample standard deviation \(s\).

Many universities and colleges have instituted supplemental instruction (SI) programs, in which a student facilitator meets regularly with a small group of students enrolled in the course to promote discussion of course material and enhance subject mastery. Suppose that students in a large statistics course (what else?) are randomly divided into a control group that will not participate in SI and a treatment group that will participate. At the end of the term, each student's total score in the course is determined. a. Are the scores from the SI group a sample from an existing population? If so, what is it? If not, what is the relevant conceptual population? b. What do you think is the advantage of randomly dividing the students into the two groups rather than letting each student choose which group to join? c. Why didn't the investigators put all students in the treatment group? Note: The article "Supplemental Instruction: An Effective Component of Student Affairs Programming" (J. of College Student Devel., 1997: 577-586) discusses the analysis of data from several SI programs.

a. For what value of \(c\) is the quantity \(\sum\left(x_{f}-c\right)^{2}\) minimized? [Hint: Take the derivative with respect to \(c\), set equal to 0 , and solve.] b. Using the result of part (a), which of the two quantities \(\Sigma\left(x_{f}-\bar{x}\right)^{2}\) and \(\Sigma\left(x_{f}-\mu\right)^{2}\) will be smaller than the other (assuming that \(\bar{x} \neq \mu)^{2}\) ?

Automated electron backscattered diffraction is now being used in the study of fracture phenomena. The following information on misorientation angle (degrees) was extracted from the article "Observations on the Faceted Initiation Site in the Dwell-Fatigue Tested Ti-6242 Alloy: Crystallographic Orientation and Size Effects (Metallurgical and Materials Trans., 2006: 1507-1518). \(\begin{array}{lcccc}\text { Class: } & 0-<5 & 5-<10 & 10-<15 & 15-<20 \\\ \text { Rel freq: } & .177 & .166 & .175 & .136 \\ \text { Class: } & 20-<30 & 30-<40 & 40-<60 & 60-<90 \\ \text { Rel freq: } & .194 & .078 & .044 & .030\end{array}\) a. Is it true that more than \(50 \%\) of the sampled angles are smaller than \(15^{\circ}\), as asserted in the paper? b. What proportion of the sampled angles are at least \(30^{\circ}\) ? c. Roughly what proportion of angles are between \(10^{\circ}\) and \(25^{\circ}\) ? d. Construct a histogram and comment on any interesting features.
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