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 39 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 219 feet. Round to 4 decimal places. c. Find the 70th percentile for the distribution of distance of fly balls. Round to 2 decimal places.

Answer 1

Given

Normally distributed

Mean 265 ft

Deviation 39 ft

Procedure

a)

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation, the z- score of a measure X is given by:

`\begin{gathered} Z=(X-\mu)/(\sigma) \\ \end{gathered}`

The Z-score measures how many standard deviations the measure is from the mean

b)

X <= 219?

`\begin{gathered} Z=(219-265)/(39) \\ Z=-1.179 \end{gathered}`

Z-score = -1.17949

P(x<219) = 0.1191

c) Find the 70th percentile

First, we have that the provided z-score is Z=0.5244.

`\begin{gathered} 0.5244=(X-265)/(39) \\ X=265+0.5244\cdot39 \\ X=285.45 \end{gathered}`