91Ó°ÊÓ

Data on the 3-point percentage, field-goal percentage, and free-throw percentage for a sample of 50 professional basketball players were obtained. Regression analyses were conducted on the relationships between 3-point percentage and field-goal percentage and between 3-point percentage and freethrow percentage. The StatCrunch results are shown below. (Source: nba.com) Simple linear regression results: Dependent Variable: 3 Point \% Independent Variable: Field Goal \% 3 Point \(\%=40.090108-0.091032596\) Field Goal \% Sample size: 50 \(\mathrm{R}\) (correlation coefficient) \(=-0.048875984\) \(\mathrm{R}-\mathrm{sq}=0.0023888618\) Estimate of error standard deviation: \(7.7329785\) Simple linear regression results: Dependent Variable: 3 Point \% Independent Variable: Free Throw \% 3 Point \(\%=-8.2347225+0.54224127\) Free Throw \(\%\) Sample size: 50 \(\mathrm{R}\) (correlation coefficient) \(=0.57040364\) \(\mathrm{R}-\mathrm{sq}=0.32536031\) Estimate of error standard deviation: \(6.3591944\) Based on this sample, is there a stronger association between 3 -point percentage and field-goal percentage or 3 -point percentage and freethrow percentage? Provide a reason for your choice based on the StatCrunch results provided.

Step by step solution

01

Analyze the Correlation Coefficient \(R\)

The correlation coefficient \(R\) for: \n\n- field-goal percentage is \(-0.048875984 \),\n- free throw percentage is \(0.57040364\). \n\nThe value of correlation coefficient \(R\) ranges from -1 to 1. The positive sign of the correlation between 3-point percentage and free throw percentage suggests a positive relationship, while the negative value of the correlation between 3-point percentage and field-goal percentage suggests a negative relationship. However, closer these values are to 0, the less of an association exists. By comparing the magnitudes (ignoring the signs for now), free throw percentage shows a stronger correlation with the 3-point percentage.

02

Analyze the Coefficient of Determination \(R^2\)

The coefficient of determination \(R^2\) for: \n\n- field-goal percentage is \(0.0023888618\),\n- free throw percentage is \(0.32536031\).\n\nThe coefficient of determination \(R^2\) ranges from 0 to 1, and tells you the percentage of dependent variable's variation that the linear model explains. Here, the free throw percentage explains \(32.53%\) (obtained by multiplying \(0.32536031\) with 100) of the variance in 3-point percentage, while the field-goal percentage accounts for only about \(0.238%\) (obtained by multiplying \(0.0023888618\) with 100). Hence, free throw percentage has a stronger association with 3-point percentage based on the \(R^2\) values.

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Key Concepts

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

Correlation Coefficient

The correlation coefficient, denoted as R, is a statistical measure that calculates the strength and direction of a linear relationship between two variables. It's a value between -1 and 1, where:

- 1 indicates a perfect positive linear correlation,
- 0 indicates no correlation,
- -1 indicates a perfect negative linear correlation.

In the given exercise, the correlation coefficient for the relationship between 3-point percentage and free throw percentage is 0.57040364, indicating a moderate positive association. In contrast, the correlation for the relationship between 3-point percentage and field-goal percentage is -0.048875984, suggesting an extremely weak negative association. Thus, players with higher free throw percentages also tend to have higher 3-point percentages, a trend not as evident with field-goal percentages.

Coefficient of Determination

The coefficient of determination, usually symbolized as R², represents the proportion of the variance for the dependent variable that's explained by the independent variable in a regression model. Values range from 0, which signals no explanatory power, to 1, which indicates perfect prediction.

In the exercise, the R² for the free throw percentage model (0.32536031) tells us that approximately 32.54% of the variation in 3-point percentage can be predicted from the free throw percentage. Conversely, the field-goal percentage model has an R² of 0.0023888618, implying it explains a negligible portion of the variance in 3-point percentage. This numerical evidence reinforces that free throw performance is a more reliable predictor of 3-point shooting than field-goal percentage.

Statistical Analysis

Statistical analysis encompasses a variety of techniques used to interpret data, make predictions, and drive decisions. In this context, it involves exploratory data analysis to understand relationships between variables and includes both descriptive statistics (like means and variances) and inferential statistics (like regression analysis).

The purpose of conducting a statistical analysis on the basketball players' shooting percentages is to quantify the strength of association between different types of shooting. The approaches shown – using R and R² – permit us to draw conclusions from the provided sample to the general population of professional basketball players. The values obtained from the exercise imply that while both free-throw percentage and field-goal percentage are related to 3-point shooting, the relationship with free-throw percentage is statistically more significant.

Regression Analysis

Regression analysis is a powerful statistical method to understand and quantify the relationship between variables. In simple linear regression, the goal is to fit a linear equation to the observed data. This equation models the expected value of the dependent variable as a linear function of one independent variable.

The simple linear regression results displayed for the basketball players' data show the equations for predicting 3-point percentage from field-goal percentage and free throw percentage. The first equation suggests that there isn't much of a linear trend between 3-point and field-goal percentages, as indicated by the near-zero (and negative) R value. However, the equation for free throw percentage presents a visible upward slope, as the positive correlation coefficient indicates a more significant predictive relationship, proven useful for coaches and analysts evaluating player performance.