Question

suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 258 feet and a standard deviation of 53 feet. use your graphing calculator to answer the following questions. write your answers in percent form. round your answers to the nearest tenth of a percent. a) if one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 204 feet? ( p(\text{fewer than } 204 \text{ feet})=) % b) if one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled more than 216 feet? ( p(\text{more than } 216 \text{ feet})=) % check answer question 13

Explanation:

Step1: Calculate z - score for part (a)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 258\), \(\sigma=53\), and \(x = 204\).

$$z=\frac{204 - 258}{53}=\frac{-54}{53}\approx - 1.02$$

Using a graphing calculator (normalcdf function: normalcdf\((-\infty,204,258,53)\) or normalcdf\((-1000,-1.02)\)), we find the probability.

Step2: Calculate z - score for part (b)

Using the z - score formula \(z=\frac{x-\mu}{\sigma}\), with \(\mu = 258\), \(\sigma = 53\), and \(x = 216\)

$$z=\frac{216-258}{53}=\frac{-42}{53}\approx - 0.79$$

Using the property \(P(X>x)=1 - P(X\leq x)\). So, \(P(X > 216)=1-\text{normalcdf}(-\infty,216,258,53)\) or \(1-\text{normalcdf}(-1000,-0.79)\)

Answer:

a) \(P(\text{fewer than }204\text{ feet})\approx15.4\%\)
b) \(P(\text{more than }216\text{ feet})\approx78.5\%\)