Problem 14 - An Introduction to Mathematical Statistics and Its Applications

Question

Two manufacturing processes are available for annealing a certain kind of copper tubing, the primary difference being in the temperature required. The critical response variable is the resulting tensile strength. To compare the methods, fifteen pieces of tubing were broken into pairs. One piece from each pair was randomly selected to be annealed at a moderate temperature, the other piece at a high temperature. The resulting tensile strengths (in tons/sq in.) are listed in the following table. Analyze these data with a Wilcoxon signed rank test. Use a two-sided alternative. Let \(\alpha=0.05\). \begin{tabular}{ccc} \hline \multicolumn{3}{c}{ Tensile Strengths (tons/sq in.) } \\ \hline \multicolumn{3}{c}{ Moderate } \\ Pair & Temperature & High \\ \hline 1 & \(16.5\) & \(16.9\) \\ 2 & \(17.6\) & \(17.2\) \\ 3 & \(16.9\) & \(17.0\) \\ 4 & \(15.8\) & \(16.1\) \\ 5 & \(18.4\) & \(18.2\) \\ 6 & \(17.5\) & \(17.7\) \\ 7 & \(17.6\) & \(17.9\) \\ 8 & \(16.1\) & \(16.0\) \\ 9 & \(16.8\) & \(17.3\) \\ 10 & \(15.8\) & \(16.1\) \\ 11 & \(16.8\) & \(16.5\) \\ 12 & \(17.3\) & \(17.6\) \\ 13 & \(18.1\) & \(18.4\) \\ 14 & \(17.9\) & \(17.2\) \\ 15 & \(16.4\) & \(16.5\) \\ \hline \end{tabular}

Step-by-Step Solution

Answer

The interpretation of the result of Wilcoxon Signed Rank Test would provide a short answer to whether there is a statistically significant difference in tensile strength of the copper tubing annealed at the two different temperature conditions.

Step 1: Sever the Differences

Calculate the differences between the high and moderate tensile strengths for each pair. This is done by subtracting the tensile strength of the piece annealed at moderate temperature from the strength of the one annealed at high temperature in each pair.

Step 2: Rank the Absolute Differences

In this step, the absolute differences calculated in the previous step are ranked in ascending order, with \(1\) for the smallest difference.

Step 3: Calculate the Test Statistic

The test statistic \(W\) for the Wilcoxon Signed-Rank Test is the lesser of the sums of the positive and negative ranks.

Step 4: Determine the Critical Value and Compare

A decision regarding the null hypothesis is made by comparing the computed test statistic with the critical value in the Wilcoxon Signed-Rank Test table, considering the provided significance level \(\alpha = 0.05\). If the test statistic \(W\) is less than or equal to the critical value, the null hypothesis is rejected.

Step 5: Interpretation

The conclusion is derived from the comparison of test statistic with the critical value. If the null hypothesis was rejected, it can be stated that there is a statistically significant difference in tensile strength of copper tubing between the two annealing processes.

Key Concepts

Tensile Strength in Materials

Tensile strength is an important characteristic in materials science and engineering. It refers to the maximum amount of tensile stress a material can withstand while being stretched or pulled before failing. Understanding tensile strength is crucial as it determines the durability and usability of a material under stress.

For applications involving materials like copper tubing, tensile strength ensures that the tubing can handle pressures and forces without breaking. Engineers and scientists often test the tensile strength to compare different manufacturing processes or materials, to pick the one that offers maximum safety and efficiency.

• Maximum stress before failure
• Determines durability under stress
• Used to compare manufacturing processes or materials

This concept is integral when comparing how different annealing processes affect the physical properties of copper tubing, as seen in the given exercise.

Annealing Process Comparison

Annealing is a heat treatment process that alters the physical and sometimes chemical properties of a material to increase its ductility and reduce its hardness. By heating a material like copper tubing to a specific temperature and then cooling it, annealing changes its structure and improves its mechanical properties.

In the context of the exercise, two different annealing processes - moderate and high temperature - were compared to see how they affect the tensile strength of copper tubing.

• Heat treatment for ductility
• Improves material properties by structural changes
• Different temperature processes can result in different tensile strengths

The comparison is crucial as it helps in selecting the optimal process that offers the best combination of strength and performance for specific applications.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (usually suggesting no effect or no difference) and an alternative hypothesis (indicating some effect or difference), and then determining whether the sample data provides sufficient evidence to reject the null hypothesis.

The Wilcoxon Signed-Rank Test, used in the exercise, is a non-parametric statistical hypothesis test. It's specifically useful for comparing two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ.

• Comparative decision making based on sample data
• Non-parametric test doesn't assume a normal distribution
• Useful for paired samples when comparing different conditions

By using hypothesis testing in this context, one can rigorously evaluate the effects of different annealing processes on tensile strength, and determine if one method significantly outperforms the other.