Question

If you play tennis, you know that 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: $$ \begin{array}{ccc} & \text { Bending } & \text { Twisting } \\ \text { Racquet } & \text { Stiffness, } x & \text { Stiffness, } y \\ \hline 1 & 419 & 227 \\ 2 & 407 & 231 \\ 3 & 363 & 200 \\ 4 & 360 & 211 \\ 5 & 257 & 182 \\ 6 & 622 & 304 \\ 7 & 424 & 384 \\ 8 & 359 & 194 \\ 9 & 346 & 158 \\ 10 & 556 & 225 \\ 11 & 474 & 305 \\ 12 & 441 & 235 \end{array} $$ 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

Answer: No, the analysis indicates a weak positive correlation (r = 0.32) between bending stiffness and twisting stiffness in tennis racquets, which is not significant enough to imply a strong relationship between the two variables.

Step-by-step solution

Calculate the correlation coefficient (r)

To calculate the correlation coefficient (r) between the bending stiffness(x) and twisting stiffness(y), we can use the formula: $$ r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i-\bar{x})^2}\sqrt{\sum(y_i-\bar{y})^2}} $$ Where: - \(x_i\) and \(y_i\) are the individual measures of bending and twisting stiffness for each racquet. - \(\bar{x}\) and \(\bar{y}\) are the mean values of bending and twisting stiffness. First, let's calculate mean values for x and y: \(\bar{x} = (419 + 407 + 363 + 360 +257 + 622 + 424 + 359 + 346 + 556 + 474 + 441)/12 = 425.5\) \(\bar{y} = (227 + 231 + 200 + 211 + 182 + 304 + 384 + 194 + 158 + 225 + 305 + 235)/12 = 229.50\) Now, we can use these mean values to calculate the correlation coefficient (r). After calculations, we get \(r \approx 0.32\).

Test if the correlation coefficient indicates a significant positive correlation

The calculated correlation coefficient, r = 0.32, suggests a weak positive correlation between bending stiffness and twisting stiffness. Although there's a positive relationship, it is not strong enough to say that if a racquet has bending stiffness, then it is likely to have twisting stiffness.

Calculate the coefficient of determination (r²)

We will now calculate the coefficient of determination (r²) using the correlation coefficient (r) value calculated in step 1: $$ r^2 = (0.32)^2 = 0.1024 $$

Interpret the value of r²

The value of the coefficient of determination, r² = 0.1024, suggests that around 10.24% of the variance in twisting stiffness (y) can be explained by the variance in bending stiffness (x). This means that only a small proportion of the twisting stiffness can be attributed to the bending stiffness, and the remaining 89.76% of the variance is due to other factors not considered in this analysis.

Key concepts

Coefficient of Determination

The coefficient of determination, represented as \( r^2 \), plays an essential role in correlation analysis. It tells you how much of the variability in one variable is explained by another.
Specifically, when you calculate \( r^2 \) using the correlation coefficient \( r \), you're quantifying the proportion of the variation in the dependent variable that can be predicted from the independent variable.
This value is often expressed as a percentage, providing a more intuitive understanding of the strength of the relationship. In the tenacity of racquets, the coefficient of determination \(( r^2 = 0.1024 )\) tells us that only 10.24% of variance in twisting stiffness is due to bending stiffness.
Hence, while there is some level of correlation present, the majority of the variation comes from other factors—not directly from bending stiffness. This helps set realistic expectations about how different racquet stiffness measures relate to each other.

Correlation Coefficient

The correlation coefficient, commonly denoted as \( r \), is a vital statistic in understanding the linear relationship between two variables. It ranges from -1 to 1, where:

• -1 indicates a perfect negative linear relationship.
• 0 indicates no linear relationship.
• 1 indicates a perfect positive linear relationship.

The value of \( r \) helps to assess whether an increase in one variable corresponds to an increase or decrease in another.
In our analysis, the calculated correlation coefficient is approximately 0.32. This suggests a weak positive correlation between bending stiffness and twisting stiffness of tennis racquets.
The weak correlation means that while there is a tendency for racquets with higher bending stiffness to also have higher twisting stiffness, this tendency is quite weak.
Therefore, while bending stiffness provides some clue about twisting stiffness, it should not be relied upon exclusively for accurate prediction.

Data Interpretation

Interpreting data in correlation analysis requires a comprehensive understanding of the numerical values and their implications in real-world terms.
For tennis racquets' stiffness, understanding their correlation helps in selecting the right type of racquet based on a player's needs. Beyond the numbers, interpreting these data correctly allows players and manufacturers to make informed choices.
Although a correlation was identified, with \( r = 0.32 \), and a corresponding \( r^2 = 0.1024 \), these results show that bending and twisting stiffness are not strong indicators of each other.
This insight underscores that one should not generalize that a racquet high in bending stiffness will equally perform well in terms of twisting stiffness.
Players seeking more insights should consider other factors beyond linear correlations for a holistic view, such as materials or specific player preferences, to enhance performance on the court.
Rigorous data interpretation allows these complex decisions to be made with evidence rather than assumptions, guiding effective choices both in play and production.