Part A

How would you describe Melvin’s overall record? Describe your prediction regarding Melvin’s attempts at field goals.

Coach Trayvon recorded a total of 952 attempted field goals by Melvin, 79.7% of which resulted in a successful goal. The majority of the data was recorded during practice (872 of the 952 field goal attempts). The minimum and maximum distances attempted were 18.0 yards and 62.0 yards, respectively. The average distance was 36.5 yards. As seen in the graph below, the likelihood of Melvin successfully scoring a field goal gradually dropped as the attempted yardage increased.

The probability of Melvin successfully scoring a field goal can be estimated using the following equation: \[ P(y=1) = \frac{e^{5.689-0.110x}}{1+e^{5.689-0.110x}} \]

The probability that Melvin would score ranged from 24.7% to 97.6%, with an average of 84.3%. Given these probabilities, Melvin was more likely to score an attempted field goal than be unsuccessful. However when the distance exceeded 51.7 yards, he was more likely to miss the goal (y = 0.5).

This model, however, is not a good fit for predicting the response variable, or the likelihood that Melvin will be successful in scoring a field goal. The null deviance of 959.93 reveals that the model is a bad fit. The addition of the independent variable does improve the fit as shown by a lower residual deviance of 821.77, but it is still indicative of poor fit.

Part B

Do Melvin’s attempts in matches differ from his performance in practice?

Melvin’s attempts in matches slightly differ from his performance in practice; he has a higher probability of scoring during practice than in a match. This could be related to a number of factors, but there is significantly more data describing his performance in practice (872 recorded points) than in matches (80 recorded points).

  1. Logistic Equations

    • Practice \[ P(y=1) = \frac{e^{5.691-0.109x}}{1+e^{5.691-0.109x}} \]
    • Matches \[ P(y=1) = \frac{e^{5.506-0.111x}}{1+e^{5.506-0.111x}} \]

  2. What is the probability of Melvin scoring a goal when he kicks from 20, 40, and 60 yards in practice?

    20 Yards 40 Yards 60 Yards

    0.9708687

    0.7895329

    0.2968897

  3. What is the probability of Melvin scoring a goal when he kicks from 20, 40, and 60 yards in matches?

    20 Yards 40 Yards 60 Yards

    0.9640803

    0.7453777

    0.2420139

Part C

Using ggplot, plot the logistic models for Melvin’s performance in matches as well as in practice in a single plot.