Question

The data shown in the accompanying table give measures of bending stiffness and twisting stiffness as determined by engineering tests on 12 tennis racquets. $$\begin{array}{ccc}\hline \multirow{2}{*} {\text { Racquet }} & \begin{array}{c}\text { Bending } \\\\\text { Stiffness }\end{array} &\begin{array}{c}\text { Twisting } \\\\\text { Stiffness }\end{array} \\\\\hline 1 & 419 & 227 \\\2 & 407 & 231 \\\3 & 363 & 200 \\\4 & 360 & 211 \\\5 & 257 & 182 \\\6 & 622 & 304 \\\7 & 424 & 384 \\\8 & 359 & 194 \\\9 & 346 & 158 \\\10 & 556 & 225 \\\11 & 474 & 305 \\\12 & 441 & 235 \\\\\hline\end{array}$$ a. Calculate the rank correlation coefficient \(r_{s}\) between bending stiffness and twisting stiffness. b. If a racquet has bending stiffness, is it also likely to have twisting stiffness? Use the rank correlation coefficient to determine whether there is a significant positive relationship between bending stiffness and twisting stiffness. Use \(\alpha=.05 .\)

Short answer

Short Answer: The rank correlation coefficient (r_s) between bending stiffness and twisting stiffness is 0.6545. Yes, a racquet with bending stiffness is likely to have twisting stiffness too, as there is a significant positive relationship between the two variables (r_s > 0.576, the critical value).

Step-by-step solution

Rank the values

Firstly, we will rank the values of both bending and twisting stiffness separately, in ascending order. Assign a rank of 1 for the smallest values and continue accordingly.

Calculate the difference between the ranks

After assigning the ranks, find the difference between the ranks of the corresponding values of bending stiffness and twisting stiffness, denoted as \(d_i\).

Calculate squared differences

Square all the differences from Step 2, and sum them up. This is referred to as the sum of the squared differences, denoted by \(\sum d_i^2\).

Calculate the rank correlation coefficient

Now, use the formula to calculate the rank correlation coefficient, \(r_s\): $$r_s = 1 - \frac{6\sum d_i^2}{n(n^2 - 1)}$$ where \(n\) is the number of data pairs (12 racquets in this case).

Determine the nature of the relationship

Compare the calculated \(r_s\) value with the critical value from a table with \(\alpha = 0.05\) for \(n = 12\), considering the rank correlation coefficient value. If \(|r_s|\) is greater than the critical value, it indicates a significant positive (or negative) relationship between bending stiffness and twisting stiffness. Let's perform the calculations to find the solution. Step 1 and 2: Ranks and differences for each racquet: | Racquet | Bending Stiffness Rank | Twisting Stiffness Rank | \(d_i\) | |---------|------------------------|-------------------------|-------| | 1 | 7 | 6 | 1 | | 2 | 6 | 7 | -1 | | 3 | 4 | 3 | 1 | | 4 | 3 | 4 | -1 | | 5 | 1 | 1 | 0 | | 6 | 12 | 11 | 1 | | 7 | 8 | 12 | -4 | | 8 | 2 | 2 | 0 | | 9 | 5 | 5 | 0 | | 10 | 11 | 8 | 3 | | 11 | 10 | 10 | 0 | | 12 | 9 | 9 | 0 | Step 3: Calculate the sum of the squared differences: \(\sum d_i^2 = 1^2 + (-1)^2 + 1^2 + (-1)^2 + 0^2 + 1^2 + (-4)^2 + 0^2 + 0^2 + 3^2 + 0^2 + 0^2 = 28\) Step 4: Calculate \(r_s\): \(r_s = 1 - \frac{6 \times 28}{12 (12^2 - 1)} = 0.6545\) Step 5: For \(n = 12\) and \(\alpha = 0.05\), the critical value from the table is approximately 0.576. Since the calculated \(r_s = 0.6545 > 0.576\), there is a significant positive relationship between bending stiffness and twisting stiffness. So, the answers are: a. The rank correlation coefficient \(r_s\) between bending stiffness and twisting stiffness is 0.6545. b. Yes, if a racquet has bending stiffness, it is likely to have twisting stiffness too, as there is a significant positive relationship between the two variables.

Key concepts

Understanding Bending Stiffness

Bending stiffness is a parameter that represents the resistance of a material or structure to bending. In the context of tennis racquets, higher bending stiffness means that the racquet is less likely to flex when it impacts the ball. This can affect the player’s control over the shot and the power they can transfer to the ball.

For engineers and product designers, quantifying bending stiffness is crucial for creating racquets that meet specific performance criteria. It's not just about making a racquet stiff; it's about optimizing stiffness for specific playing characteristics. Players may have preferences depending on their style of play—some may prefer a stiffer racquet for more power, while others might want more flexibility for better control.

To measure the bending stiffness of tennis racquets, as in the exercise, engineers apply a force and measure the degree of bending. This data, when compared across various racquets, can inform both manufacturers and consumers about which racquet might suit their needs best.

Twisting Stiffness Explained

Twisting stiffness, or torsional stiffness, in contrast to bending stiffness, describes a racquet’s resistance to twisting forces. When a tennis ball strikes the racquet off-center, a moment or torque is generated, causing the racquet to twist. This can result in less control over the ball direction and a potential loss of power.

The ability to resist such twisting is essential for precision in a game where every small angle can alter the ball’s trajectory dramatically. A racquet with high twisting stiffness will maintain stability upon impact, particularly for shots that hit near the tip or the sides of the racquet head.

During the testing process for tennis racquets, engineers measure how much a racquet twists under a given torque. Higher values indicate a racquet with greater twisting stiffness, which often correlates to improved performance for advanced players who hit the ball with high intensity or at extreme angles.

Statistical Significance

Statistical significance is a term used to ascertain if the observed relationships or differences in data are due to chance or if they reflect true effects. In the context of the racquet study, the calculation of the rank correlation coefficient helps determine if there's a genuine link between bending and twisting stiffness.

By setting a significance level, commonly denoted as \(\alpha\), researchers decide on a threshold for which the results would be considered statistically significant. In our case, with \(\alpha = 0.05\), we are indicating that we would only expect the observed correlation to be a product of random variation 5% of the time or less.

If the computed rank correlation coefficient exceeds the critical value associated with the chosen \(\alpha\), we can confidently state that the relationship between bending and twisting stiffness is significant and not due to random chance. This statistical assurance allows manufacturers to make informed decisions based on the analysis.

Non-parametric Tests

Non-parametric tests are statistical methods used when data doesn't adhere to the common assumptions required for traditional parametric tests. One such assumption is that the data follows a normal distribution. In many real-world scenarios, including the tennis racquets' stiffness measurements, these conditions are not met, necessitating an alternative testing method.

The rank correlation coefficient, calculated in the exercise, is part of a non-parametric test called Spearman's rank-order correlation. This test evaluates the degree of association between two variables by solely considering the ranks rather than the actual data values. It's particularly useful when examining ordinal data, where the exact differences between levels are not clear or when dealing with outliers that would skew the results of parametric tests.

Use of non-parametric tests ensures that conclusions drawn from statistical analyses are robust and reliable, even when the data behaves unpredictably. Conducting such tests is essential for accurately understanding relationships in datasets across various fields, including sports engineering.