Variability in Baseball Suppose, during the course of a typical season, a batter has 500 at-bats. This means the player has the opportunity to get a hit 500 times during the course of a season. Further, suppose a batter is a career 0.280 hitter (he averages 280 hits every 1000 at-bats or he has 280 successes in 1000 trials of the experiment), so the population proportion of hits is 0.280.


b. Would it be unusual for a player who is a career 0.280 hitter to have a season in which he hits at least 0.310?

Verified step by step guidance

1. Identify the problem as a question about whether a particular observed batting average (0.310) is unusual for a player whose true batting average (population proportion) is 0.280 over 500 at-bats. This involves understanding the variability of the sample proportion around the population proportion.
2. Define the parameters: the number of trials (at-bats) \(n = 500\), the population proportion of success (hits) \(p = 0.280\), and the observed sample proportion \(\hat{p} = 0.310\).
3. Calculate the standard deviation (standard error) of the sample proportion using the formula for the standard deviation of a binomial proportion: \[ \text{SE} = \sqrt{\frac{p(1-p)}{n}} \]
4. Compute the z-score to measure how many standard errors the observed sample proportion is away from the population proportion: \[ z = \frac{\hat{p} - p}{\text{SE}} \]
5. Interpret the z-score by comparing it to common critical values (e.g., 1.96 for 95% confidence). If the z-score is large (typically greater than 2 or less than -2), the observed batting average would be considered unusual. Otherwise, it is within the expected variability.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. In this context, each at-bat is a trial with a success probability of 0.280 (getting a hit). Understanding this distribution helps calculate the likelihood of different numbers of hits in a season.

Normal Approximation to the Binomial

When the number of trials is large, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1-p). This simplifies calculations of probabilities, such as the chance of hitting at least 0.310 in a season, by using z-scores and standard normal tables.

Unusual Events and Probability Thresholds

An event is considered unusual if its probability is very low, often below 5%. To determine if hitting at least 0.310 is unusual, we calculate the probability of this outcome and compare it to the threshold. This concept helps interpret whether a season’s performance significantly deviates from the player's typical ability.