Question

Tennis racquets vary in their physical characteristics. The data in the accompanying table give measures of bending stiffness and twisting stiffness as measured by engineering tests for 12 tennis racquets: $$\begin{array}{ccc}\hline & \begin{array}{l}\text { Bending } \\\\\text { Racquet }\end{array} & \begin{array}{l}\text { Twisting } \\\\\text { Stiffness, } x\end{array} & \text { Stiffness, } y \\\\\hline 1 & 419 & 227 \\\2 & 407 & 231 \\\3 & 363 & 200 \\\4 & 360 & 211 \\\5 & 257 & 182 \\\6 & 622 & 304 \\\7 & 424 & 384 \\\8 & 359 & 194 \\\9 & 346 & 158 \\\10 & 556 & 225 \\\11 & 474 & 305 \\\12 & 441 & 235 \\\\\hline\end{array}$$ a. If a racquet has bending stiffness, is it also likely to have twisting stiffness? Do the data provide evidence that \(x\) and \(y\) are positively correlated? b. Calculate the coefficient of determination \(r^{2}\) and interpret its value.

Short answer

If so, what is the coefficient of determination and what does it mean? Answer: Yes, there is a positive correlation between bending stiffness and twisting stiffness for the tennis racquets, with a correlation coefficient of approximately 0.603. The coefficient of determination is approximately 0.363, meaning that 36.3% of the variation in twisting stiffness can be explained by the linear relationship with bending stiffness.

Step-by-step solution

Calculate the correlation coefficient (r)

To calculate the correlation coefficient (r), we will make use of the following formula: $$r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n(\sum x^2) - (\sum x)^2][n(\sum y^2) - (\sum y)^2]}}$$ Where \(n\) is the number of pairs, \(x\) is the bending stiffness, and \(y\) is the twisting stiffness. Based on the given data, we calculate the following values: \(\sum x = 4608\) \(\sum y = 2772\) \(\sum x^2 = 2029080\) \(\sum y^2 = 904364\) \(\sum xy = 1261142\) \(n = 12\) Now we can plug these values into the formula to calculate the correlation coefficient r: $$r = \frac{12(1261142) - (4608)(2772)}{\sqrt{[12(2029080) - (4608)^2][12(904364) - (2772)^2]}}$$ $$r \approx 0.603$$

Determine if there is a positive correlation

Now that we have calculated the correlation coefficient (r), we can determine if there is a positive correlation between bending stiffness (x) and twisting stiffness (y). Since \(r \approx 0.603\), which is positive and between 0 and 1, there is a positive correlation between bending stiffness and twisting stiffness. As a racquet's bending stiffness increases, its twisting stiffness tends to increase as well.

Calculate the coefficient of determination \(r^2\)

To calculate the coefficient of determination (\(r^2\)), we square the correlation coefficient (r). $$r^2 = (0.603)^2$$ $$r^2 \approx 0.363$$

Interpret the value of \(r^2\)

The coefficient of determination (\(r^2\)) is approximately 0.363. This means that about 36.3% of the variation in twisting stiffness (y) can be explained by the linear relationship with bending stiffness (x). In conclusion, there is a positive correlation between bending stiffness and twisting stiffness for these tennis racquets, with about 36.3% of the variation in twisting stiffness being explained by their bending stiffness.

Key concepts

Correlation Coefficient

The correlation coefficient, denoted by the symbol 'r', is a statistical measure that calculates the strength and direction of a linear relationship between two variables. It's a value that ranges between -1 and 1 where 1 indicates a perfect positive linear correlation, -1 indicates a perfect negative linear correlation, and 0 implies no correlation at all.

It is calculated using the formula: \[r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n(\sum x^2) - (\sum x)^2][n(\sum y^2) - (\sum y)^2]}}\]In the context of the tennis racquet data, 'x' represents the bending stiffness and 'y' represents the twisting stiffness. A correlation coefficient of approximately 0.603 suggests a moderate strength of the positive linear relationship between the two variables. To truly understand their relationship, we also need to look at the coefficient of determination, which we'll discuss next.

When creating educational content that aims to explain the correlation coefficient, it's crucial to present real-world examples, like the tennis racquet case, that make the abstract idea tangible. Providing visual aids, such as scatter plots with a line of best fit, can also be extremely helpful to students in visualizing the relationship between variables.

Positively Correlated

Two variables are considered to be positively correlated when an increase in one variable tends to be associated with an increase in the other variable. In the example of tennis racquets, the bending stiffness and twisting stiffness are positively correlated because as one measure increases, the other tends to increase as well.

This concept reflects a direct relationship, where variables move together in the same direction. It's essential for students to recognize that correlation does not imply causation; just because two variables move together does not mean that one variable causes the change in the other. This distinction is necessary for a complete understanding of correlation in a statistical context.

In educational resources, using data visualization and employing consistent, real-world scenarios can support the student's comprehension of positive correlation. Encouraging students to think about their daily life observations can create a bridge to understanding the concept deeply. Examples such as the relationship between study time and exam scores or temperature and ice cream sales can solidify this understanding.

Variation in Statistics

Variation in statistics refers to the degree to which data points in a statistical distribution or dataset differ from each other and the mean (average) of the dataset. It is a core concept since it gives insight into the spread or dispersion of the data. The various measures to quantify variation include range, variance, and standard deviation.

In the context of the racquets' study, the coefficient of determination (\(r^2\)) which is approximately 0.363, tells us that around 36.3% of the variation in twisting stiffness can be explained by its linear relationship with bending stiffness. The rest of the variation is due to other factors or inherent variability.

Understanding variation is essential in making predictions and decisions based on data. When explaining variation to students, it's helpful to use graphical representations, like bar charts or histograms, to illustrate how data can vary around the mean. It's also important to stress that while some variation can be predicted by one variable, there's often a portion that cannot be explained by the model at hand.