The batting average of a baseball player is the number of “hits” divided by the number of “at-bats.” Recently, a certain major league player’s at-bats and corresponding hits were recorded for 200 consecutive games. The consecutive games span more than one season. Since each game is different, the number of at-bats and hits both vary. For this particular player, there were from zero to five at-bats. Thus, one can sort the 200 games into six categories:
Consider the games where the player had exactly four at-bats. A similar analysis can be done for each of the other at-bats category. Download the file titled Bats. It contains a scatter plot of the four at-bats number of hits versus frequency. To compare the results to the Binomial Distribution, complete the following:
Explain why the four at-bats is a binomial experiment.
Using the Bats scatter plot, construct a frequency distribution for the number of hits.
Compute the mean number of hits. The formula for the mean is (added as a file) .
Here, xi represent no. of hits (0, 1, 2, 3, 4) and fi is the corresponding frequency. Explain what the numerical result means.
From the frequency distribution, construct the corresponding probability distribution. Explain why it is a probability distribution. Then, use Excel to make a scatter plot of the probability distribution:
Select the two columns of the probability distribution. Click on INSERT, and then go to the Charts area and select Scatter. Then choose the first Scatter chart (the one without lines connecting).
Using the frequency distribution, what is the player’s batting average for four at-bats? In part 3, note that the numerator in the formula for the mean is the total number of hits. The total number of at-bats is the denominator of the formula for the mean multiplied by 4.
The Binomial Distribution is uniquely determined by n, the number of trials, and p, the probability of “success” on each trial. Using Excel, construct the Binomial Probability Distribution for four trials, n, and probability of success, p, as the batting average in part 5. Here is an explanation of the BINOM.DIST function (Links to an external site.)Links to an external site. in Excel.
For example, In Excel
represents the probability of 7 successes out of 15 (n) trials. The 0.7 is the probability of success, p.
Using the formula for the mean of the binomial distribution, what is the mean number of successes in part 6 up above?
In Excel, make a scatter plot for the binomial distribution. The instructions for making one are in part 4 up above.
Use the results up above to compare the probability distribution of four at bats and the Binomial Distribution. Compare the means in parts 4 and 6, too. If the probability distribution of 4 at bats and the Binomial Distribution differ, explain why that is so.
Write a report that adheres to the Written Assignment Requirements.
Submit your Excel file in addition to your report.
Paper must be written in third person.
Your paper should be four to five pages in length (counting the title page and references page) and cite and integrate at least two credible outside sources.
Include a title page, introduction, body, conclusion, and a reference page.
The introduction should describe or summarize the topic or problem. It might discuss the importance of the topic or how it affects you or society as a whole, or it might discuss or describe the unique terminology associated with the topic.
The body of your paper should answer the questions posed in the problem. Explain how you approached and answered the question or solved the problem, and, for each question, show all steps involved. Be sure this is in paragraph format, not numbered answers like a homework assignment.
The conclusion should summarize your thoughts about what you have determined from the data and your analysis, often with a broader personal or societal perspective in mind. Nothing new should be introduced in the conclusion that was not previously discussed in the body paragraphs.
Include any tables of data or calculations, calculated values, and/or graphs associated with this problem in the body of your assignment.
A random variable X follows a binomial distribution if its probability space consists of probabilities of two possible outcomes of a phenomenon – success or failure (DeGroot & Schervish, 2012). A binomial experiment describes experiment consisting of n repeated trials, with outcomes that can be defined as either success or failure. Further, the distribution is defined by two significant properties, number of trials (n) and probability of success (p). This paper is an analytical approach of the fit of binomial distribution in a base-ball player’s at-bat results from results obtained from his baseball games.
This is a binomial experiment because in every at bat, there are two possible outcomes in the sample space – either hit or out. In this case, success is defined by the player hitting the base, an out defines failure.
Number of Hits
Table 1: Frequency table of the number of hits in games.
The mean of the distribution is 1.088. This means that it is expected that the player will be getting above 1 at-bat in each game averagely.
This is a probability distribution because it covers the entire probability space, hence can be defined along the underlying sample space whose probability P(S)=1. Moreover, since the model here is a set of phenomenal outcomes (0, 1, 2, 3, 4), making it a discrete probability distribution (Chen, 2010). The specific probabilities developed from frequencies provide the probabilities of outcomes/occurrences of the phenomenon (at-bats). In this scenario of fixed four at bats, the probability distribution in case that the player has the highest chance of hitting one at-bat than he does all the other possible outcomes. This defines the probability space based on the sampled games with number of hits with four at-bats.
Fig 1: Probability Distribution plot.
The batting average for four bats is
In calculation of the binomial probability distribution, the probability for 4 bats is fixed at 0.272
Therefore, the binomial distribution is determined by parameters n = 4 and p = 0.272.
Fig 2: Scatter of the Binomial Probability Distribution
Mean = np
Though the means are relatively equal, the probability distribution and the binomial distribution differ significantly, based on the player’s outcome at four at bats. For 0, 3 and 4 hits, the player’s outcome (in the probability distribution) is higher than the expected (in the binomial distribution). The player also had more 1-hit and 2-hit than expected. This does not mean that binomial distribution formula is a poor fit for the player’s at-bat results, as also argued out by Abelson (2012).
Therefore, though binomial distribution shows significant variations with the probability distribution of the observed data of the player, it fits well in estimating the player’s bat results.
Abelson, R. P. (2012). Statistics as principled argument. Psychology Press.
Chen, Y. (2010). Introduction to probability theory. Lecture Notes on Information Theory, Duisburg-Essen Univ., Duisburg, Germany.
DeGroot, M. H., & Schervish, M. J. (2012). Probability and statistics. Pearson Education.