Question

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet.

If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? (Round to 4 decimal places)

Answer 1

To find the probability that a fly ball travels fewer than 220 feet in a normally distributed setting, we need to standardize the score using the z-score formula.

The formula for the z-score is:

\[
z = \frac{(X - \mu)}{\sigma}
\]

where:
- \(X\) is the value for which we are finding the probability (220 feet),
- \(\mu\) is the mean (250 feet),
- \(\sigma\) is the standard deviation (50 feet).

Substituting the values into the formula:

\[
z = \frac{(220 - 250)}{50} = \frac{-30}{50} = -0.6
\]

Now we need to find the probability that \(Z\) (the standard normal variable) is less than -0.6. This can be found using the standard normal distribution table or a calculator.

Looking up the z-value of -0.6, we find:

\[
P(Z < -0.6) \approx 0.2743
\]

Thus, the probability that a randomly chosen fly ball traveled fewer than 220 feet is approximately:

\[
\boxed{0.2743}
\]