Question

suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 265 feet and a standard deviation of 35 feet. let x be the distance in feet for a fly ball.
a. what is the distribution of x? x ~ n(□, □)
b. find the probability that a randomly hit fly ball travels less than 268 feet. round to 4 decimal places.
□
c. find the 85th percentile for the distribution of distance of fly balls. round to 2 decimal places.
□ feet

Explanation:

Step1: Identify normal distribution parameters

Given $X \sim N(\mu, \sigma^2)$, $\mu=265$, $\sigma=35$ so $\sigma^2=1225$.

Step2: Calculate z-score for 268 feet

$z = \frac{X - \mu}{\sigma} = \frac{268 - 265}{35} \approx 0.0857$.

Step3: Find probability for z=0.0857

Use z-table: $P(Z < 0.0857) \approx 0.5341$.

Step4: Find z-score for 85th percentile

From z-table, $z \approx 1.036$ (since $P(Z < 1.036) \approx 0.85$).

Step5: Calculate 85th percentile value

$X = \mu + z\sigma = 265 + 1.036 \times 35 \approx 265 + 36.26 = 301.26$.

Answer:

a. 265, 1225
b. 0.5341
c. 301.26 feet