a binomial distribution will be approximately correct as a model for one of these two sports settings and for the other. explain why by briefly discussing both settings.
a)a national football league kicker has made 80% of his field goal attempts in the past. this season he attempts 20 field goals.the attempts differ widely in distance angle wind an so on..
b) a basketball player has made 80% of his free throws in the past. this season he takes 150 free throws. free throws are attempted from 15 feet away with no interference.
To determine whether a binomial distribution is an appropriate model for each of the two sports settings, let's evaluate them individually:
a) For the NFL kicker attempting field goals, a binomial distribution is not the most accurate model. This is because the attempts differ widely in terms of distance, angle, wind, and other factors. These varying conditions introduce complexities that go beyond the scope of a simple binomial distribution. In this scenario, the outcome of each individual field goal attempt depends on multiple factors, making it difficult to model accurately with a binomial distribution.
b) On the other hand, for the basketball player attempting free throws, a binomial distribution is a reasonable approximation. Free throws are typically attempted from the same distance (15 feet) with no interference, which creates a more consistent shooting environment. The outcome of each free throw depends primarily on the player's shooting ability, with fewer variables affecting the outcome compared to the NFL kicker scenario. Therefore, a binomial distribution can provide a reasonably accurate representation of the basketball player's free throw performance.
In summary, a binomial distribution will be approximately correct as a model for the basketball player's free throw attempts (setting b) due to the relatively consistent shooting conditions. However, it is not an ideal model for the NFL kicker's field goal attempts (setting a) due to the wide range of factors impacting the outcome of each kick.
a) The binomial distribution is approximately correct as a model for the National Football League kicker's field goal attempts. This is because the attempts differ widely in variables such as distance, angle, wind, and other external factors. Each field goal attempt can be considered as a independent trial with a fixed probability of success (i.e., making the field goal) and failure (i.e., missing the field goal). The outcome of each attempt is not affected by the results of previous attempts, and the probability of success remains constant at 80%. The total number of successful attempts out of the 20 field goals can be modeled using a binomial distribution.
b) The binomial distribution is not an appropriate model for the basketball player's free throws. Unlike the football field goals, free throws in basketball are attempted from a fixed distance (15 feet) with no interference. Each free throw can be considered as an independent trial with a fixed probability of success (i.e., making the free throw) and failure (i.e., missing the free throw). However, the key difference is that the probability of success in this case remains constant at 80%, but the number of free throws attempted (150) is large relative to the usual assumptions of the binomial distribution. When the number of trials is large and the probability of success is not small or large, the binomial distribution can be approximated by the normal distribution. Therefore, for the basketball player's free throws, a normal distribution would be a more appropriate model.
There are only two possible scenarios with binomial distribution problems. In other words, something did or did not take place. Consider the other factors in both problems to determine the best bet for a binomial distribution.
I hope this will help.