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.
d. Explain why a career 0.280 hitter could easily have a batting average between 0.260 and 0.300.

Verified step by step guidance

1. Understand that the batter's hitting performance over 500 at-bats can be modeled as a binomial distribution with parameters \(n = 500\) (number of trials) and \(p = 0.280\) (probability of success, i.e., getting a hit).
2. Calculate the expected number of hits, which is the mean of the binomial distribution: \(\mu = n \times p = 500 \times 0.280\).
3. Calculate the standard deviation of the number of hits using the formula for a binomial distribution: \(\sigma = \sqrt{n \times p \times (1 - p)} = \sqrt{500 \times 0.280 \times 0.720}\).
4. Convert the number of hits to a batting average by dividing by the number of at-bats, so the mean batting average is \(p = 0.280\) and the standard deviation of the batting average is \(\sigma_{avg} = \frac{\sigma}{n} = \frac{\sqrt{n \times p \times (1 - p)}}{n}\).
5. Explain that because of this variability (standard deviation), the batter's observed batting average in a season can fluctuate around the true average of 0.280, making it quite common to see averages between 0.260 and 0.300 due to natural random variation in performance.

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). This distribution helps explain the variability in the number of hits a batter can have over 500 at-bats.

Sampling Variability

Sampling variability refers to the natural fluctuations in sample outcomes due to chance, even when the underlying probability remains constant. For a batter with a true average of 0.280, the observed batting average in a season can vary around this value, sometimes falling between 0.260 and 0.300 simply due to random variation.

Standard Error and Confidence Intervals

The standard error measures the expected variability of a sample proportion from the true population proportion. Using the standard error, we can construct confidence intervals to estimate the range in which the batter's average is likely to fall. This explains why a 0.280 hitter might have a batting average between 0.260 and 0.300 in a given season.