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Torsion during hip external rotation and extension may explain why acetabular labral tears occur in professional athletes. The article "Hip Rotational Velocities During the Full Golf Swing" \((J\). Sport Sci. Med., 2009: 296 - 299) reported on an investigation in which lead hip internal peak rotational velocity \((x)\) and trailing hip peak external rotational velocity \((y)\) were determined for a sample of 15 golfers. Data provided by the article's authors was used to calculate the following summary quantities: $$ \begin{aligned} &S_{x x}=64,732.83, \quad S_{y y}=130,566.96, \\ &S_{x y}=44,185.87 \end{aligned} $$ Separate normal probability plots showed very substantial linear patterns. a. Calculate a point estimate for the population correlation coefficient. b. If the simple linear regression model were fit to the data, what proportion of variation in external velocity could be attributed to the model relationship? What would happen to this proportion if the roles of \(x\) and \(y\) were reversed? Explain. c. Carry out a test at significance level .01 to decide whether there is a linear relationship between the two velocities in the sampled population; your conclusion should be based on a \(P\)-value. d. Would the conclusion of (c) have changed if you had tested appropriate hypotheses to decide whether there is a positive linear association in the population? What if a significance level of \(.05\) rather than \(.01\) had been used?

Step by step solution

01

Calculate the point estimate for the correlation coefficient

The sample correlation coefficient \( r \) is calculated using the formula:\[ r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}\]Substitute the given values:\[ r = \frac{44,185.87}{\sqrt{64,732.83 \cdot 130,566.96}} \]Calculate the values inside the square root and then divide to find \( r \).

02

Calculate proportion of variation explained

The proportion of variation in \( y \) explained by \( x \) is the square of the correlation coefficient, \( r^2 \). Use the \( r \) calculated in Step 1 to find this value.

03

Reversing roles of x and y

Reversing the roles of \( x \) and \( y \) does not change the correlation coefficient \( r \), so the proportion \( r^2 \) remains the same whether predicting \( y \) from \( x \) or \( x \) from \( y \).

04

Hypothesis test for correlation at 0.01 significance

To test if there is a linear relationship, use the hypotheses:- Null Hypothesis (\( H_0 \)): \( \rho = 0 \) (no linear relationship)- Alternative Hypothesis (\( H_a \)): \( \rho eq 0 \) (there is a linear relationship)Calculate the t-statistic:\[t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\]Assuming \( n = 15 \), then find the p-value for this \( t \)-statistic. Compare it against the 0.01 significance level.

05

Conclusion based on two-tailed test

If the p-value is less than 0.01, reject the null hypothesis; otherwise, fail to reject it. This determines if there is a statistically significant linear relationship.

06

Consideration of positive correlation and different significance level

For testing positive correlation, use a one-tailed test. The results might change if the alternative hypothesis becomes \( \rho > 0 \). The lower significance level of 0.05 would make it easier to find a significant result, possibly altering the decision compared to a 0.01 level.

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Key Concepts

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

Linear Relationship

A linear relationship between two variables implies that there is a consistent, proportional relationship between them. In the context of the correlation coefficient, this means that as one variable increases, the other variable tends to either increase or decrease at a constant rate.

A key characteristic of a linear relationship is that it can be represented graphically as a straight line on a scatter plot. The strength and direction of this relationship are quantified by the correlation coefficient, denoted by \( r \).

When working with data, identifying a linear pattern is often the first step in understanding the connection between two variables. This helps in predicting one variable based on the other, often leading to applications like creating regression models.

Hypothesis Testing

Hypothesis testing is a statistical method that helps us make decisions about a population based on sample data. In the context of this problem, we perform hypothesis testing to determine if there is a significant linear relationship between two types of hip rotational velocities in golfers.

The process begins by setting up two hypotheses:

• The Null Hypothesis (\( H_0 \)): There is no linear relationship between the variables (\( \rho = 0 \)).
• The Alternative Hypothesis (\( H_a \)): There is a linear relationship (\( \rho eq 0 \)).

A statistical test, such as a t-test, is done to calculate a test statistic and a p-value, which helps decide whether to reject the null hypothesis. The p-value tells us the probability of observing the data if the null hypothesis is true.

If the p-value is less than the chosen level of significance, like 0.01 or 0.05, we reject the null hypothesis and conclude that a significant linear relationship exists. Otherwise, we fail to reject the null hypothesis, implying insufficient evidence to claim a relationship.

Regression Model

A regression model is a mathematical representation that describes the relationship between a dependent variable and one or more independent variables. In this context, we are focusing on a simple linear regression model, where we try to express the leading hip internal rotation as a predictor for trailing hip rotation.

The regression model is often written in the form: \[y = a + bx\]where \( y \) is the dependent variable, \( x \) is the independent variable, \( a \) is the y-intercept, and \( b \) is the slope of the line. The slope indicates how much \( y \) changes for a unit change in \( x \).

Creating a regression model involves fitting a line to a set of data points that minimizes the difference between the observed data and the line itself. This difference is known as the residual. Once a model is fit, it can be used to predict values, assess trends, and even test new hypotheses about related phenomena.

Golf Swing Analysis

Golf swing analysis uses various methods, including statistical models, to understand the mechanics of a golfer's swing. The focus here is on examining hip rotational velocities and understanding how they contribute to performance and potential injuries.

By analyzing these velocities, researchers can identify patterns that may predict outcomes such as injury risk or success in golf. This analysis is done through collecting data, calculating statistical measures like means, variances, and correlation coefficients, and fitting regression models to explore relationships between different swing components.

Understanding the correlations between different aspects of the golf swing can inform training regimes and injury prevention strategies, helping golfers optimize their performance and maintain their physical health. This is why studies and data analysis like these are invaluable in professional athletics.