91Ó°ÊÓ

The efficiency ratio for a steel specimen immersed in a phosphating tank is the weight of the phosphate coating divided by the metal loss (both in \(\mathrm{mg} / \mathrm{ft}^{2}\) ). The article "Statistical Process Control of a Phosphate Coating Line" (Wire J. Internat., May 1997: 78-81) gave the accompanying data on tank temperature \((x)\) and efficiency ratio \((y)\). $$ \begin{array}{lcclllll} \text { Temp. } & 170 & 172 & 173 & 174 & 174 & 175 & 176 \\ \text { Ratio } & .84 & 1.31 & 1.42 & 1.03 & 1.07 & 1.08 & 1.04 \\ \text { Temp. } & 177 & 180 & 180 & 180 & 180 & 180 & 181 \\ \text { Ratio } & 1.80 & 1.45 & 1.60 & 1.61 & 2.13 & 2.15 & .84 \\ \text { Temp. } & 181 & 182 & 182 & 182 & 182 & 184 & 184 \\ \text { Ratio } & 1.43 & .90 & 1.81 & 1.94 & 2.68 & 1.49 & 2.52 \\ \text { Temp. } & 185 & 186 & 188 & & & & \\ \text { Ratio } & 3.00 & 1.87 & 3.08 & & & & \end{array} $$ a. Construct stem-and-leaf displays of both temperature and efficiency ratio, and comment on interesting features. b. Is the value of efficiency ratio completely and uniquely determined by tank temperature? Explain your reasoning. c. Construct a scatter plot of the data. Does it appear that efficiency ratio could be very well predicted by the value of temperature? Explain your reasoning.

Step by step solution

01

Create Stem-and-Leaf Display for Temperature

First, we consider the temperature data: 170, 172, 173, 174, 174, 175, 176, 177, 180, 180, 180, 180, 180, 181, 181, 182, 182, 182, 182, 184, 184, 185, 186, 188. To create the stem-and-leaf display, divide the numbers into two parts: the "stem" consisting of the tens digits, and the "leaf" consisting of the units digit. **Stems**: 17, 18 **Leaves for each stem**: - 17 | 0 2 3 4 4 5 6 7 - 18 | 0 0 0 0 0 1 1 2 2 2 2 4 4 5 6 8 This display shows that temperatures mostly fall between the twenties with a larger frequency towards the higher end of 180s.

02

Create Stem-and-Leaf Display for Efficiency Ratio

Next, we look at the efficiency ratio data: 0.84, 1.31, 1.42, 1.03, 1.07, 1.08, 1.04, 1.80, 1.45, 1.60, 1.61, 2.13, 2.15, 0.84, 1.43, 0.90, 1.81, 1.94, 2.68, 1.49, 2.52, 3.00, 1.87, 3.08. **Stems**: 0, 1, 2, 3 **Leaves for each stem**: - 0 | 8 8 9 - 1 | 0 0 3 4 4 5 6 6 8 8 9 - 2 | 1 1 5 6 - 3 | 0 0 8 The efficiency ratio data shows more frequent data points between 1 and 2, with a clever peak around 1 to 1.6.

03

Analyze the Relationship between Temperature and Efficiency Ratio

By analyzing the stems for each variable, it's apparent that temperatures do not uniquely determine efficiency ratios. There are repeated temperature values with different efficiency ratios, such as several instances of 180°F with varying ratios (1.45, 1.60, 1.61, 2.13, 2.15).

04

Construct a Scatter Plot of Temperature vs. Efficiency Ratio

On the y-axis, we plot the efficiency ratio, and on the x-axis, we plot temperature. By plotting each pair (x, y), we visualize the relationship: - Observations with temperatures around 180°F show quite some variation in efficiency ratios. The scatter plot doesn't show a perfect linear relationship, indicating the efficiency ratio can't be perfectly predicted by temperature alone. However, some patterns, like higher temperatures generally resulting in higher ratios, are visible.

05

Conclusion on Predictability of Efficiency Ratio

The scatter plot shows that while there is some correlation between temperature and efficiency ratio, temperature is not the sole or perfect predictor of efficiency ratio due to variability at the same temperature points. Multiple efficiency ratios are observed for single temperatures.

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Key Concepts

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

Efficiency Ratio

The efficiency ratio in statistical process control refers to a specific measurement used to assess the performance of a process. In this context, it is defined as the weight of the phosphate coating divided by the metal loss, both measured in units of milligrams per square foot \( \mathrm{mg} / \mathrm{ft}^2 \). This ratio helps in understanding the efficiency of the phosphating process on a steel specimen.

Statistical Process Control (SPC) often uses the efficiency ratio to determine how effectively a process converts input materials into useful products. In simpler terms, it measures how well the process is working. An efficiency ratio can highlight areas of improvement and reduce waste, leading to increased productivity

• A higher efficiency ratio suggests better performance as more phosphate coating is achieved for a given amount of metal loss.
• A lower ratio might indicate inefficiencies or issues like excessive metal loss during the phosphating process.

Understanding efficiency ratios is crucial for process engineers to optimize operations, ensure quality, and drive cost-effective performance.

Scatter Plot

A scatter plot is a graphical representation used to observe the relationship between two quantitative variables. In our exercise, this involves plotting the tank temperature on the x-axis and the efficiency ratio on the y-axis.

The benefit of using a scatter plot is its ability to reveal correlations, trends, or clusters easily. Here's how you can interpret the scatter plot related to this exercise:

• Each dot represents an observation, with its position determined by the pair of values: temperature and efficiency ratio.
• If the dots loosely form a line from bottom left to top right, it suggests a positive correlation, meaning as temperature increases, the efficiency ratio tends to increase.

In this specific case, the scatter plot indicated that there wasn't a perfect correlation between the temperature and the efficiency ratio. Multiple temperatures appeared with varying efficiency ratios, suggesting other factors might influence the ratio apart from temperature.

Thus, while temperature impacts the efficiency ratio, it isn't the sole determinant. Other variables or environmental factors might be affecting the efficiency of the phosphating process.

Stem-and-Leaf Display

A stem-and-leaf display is a useful tool in displaying quantitative data and provides a visual way to observe distribution, shape, and outliers in a dataset. In this exercise, creating stem-and-leaf displays for both temperature and efficiency ratios enables visual comparison.

For the temperature data:

• The 'stem' represents the tens digit of the temperatures (i.e., 17, 18).
• The 'leaf' signifies the unit digit (i.e., 0, 2, 3).
• This visual format shows a concentration around a particular range (180s) without losing individual data points.

For the efficiency ratio:

• The stem represents whole numbers (i.e., 0, 1, 2, 3) of the efficiency ratio values.
• The leaves indicate the decimal components (i.e., .84, .90).
• This straightforward layout illustrates the frequency and concentration of the ratios, which predominantly lie between 1 and 2.

Stem-and-leaf displays allow for quick analysis by providing both a numerical summary and a visual gauge, enabling students to recognize patterns, gaps, and concentration areas effectively.