Final answer:
a. The proportion of players with bats longer than 38 inches is approximately 33.4%. b. The top 75% of players use bats longer than 44.056 inches. c. Approximately 76.06% of bats will be allowed for use by players.
Step-by-step explanation:
a. To find the proportion of players with bats longer than 38 inches, we need to calculate the z-score for 38 inches using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. The formula becomes z = (38 - 40.5) / 5.8 = -0.431. We can then use a standard normal distribution table or a calculator to find the corresponding proportion, which is approximately 0.334 (or 33.4%).
b. To find the length of bats used by the top 75% of players, we need to calculate the z-score corresponding to the 75th percentile, which is 0.674. Rearranging the z-score formula, we get x = μ + (z * σ). Plugging in the values, x = 40.5 + (0.674 * 5.8) = 44.056 inches.
c. To find the proportion of bats allowed by the new MLB guidelines, we need to find the proportions of bats shorter than 32 inches and longer than 42 inches and subtract those from 1 (to find the complement). Using the z-score formula, we find the z-score for 32 inches is -1.71 and for 42 inches is 0.86. Using a standard normal distribution table or a calculator, we find the proportion of bats shorter than 32 inches is approximately 0.0445 and the proportion of bats longer than 42 inches is approximately 0.1949. The complement is 1 - 0.0445 - 0.1949 = 0.7606 (or 76.06%).