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. We randomly sample 49 fly balls.

a. If $X$the average distance in feet for 49 fly balls, then $\overline{X}~$

b. What is the probability that the 49 balls traveled an average of fewer than 240 feet? Sketch the graph. Scale the horizontal axis for$\overline{X}$. Shade the region corresponding to the probability. Find the probability.

c. Find the $80th$percentile of the distribution of the average of 49 fly balls.

Short answer

1. $X~N(250,7.143)$
   
2. The probability that the $49$balls traveled an average of fewer than $240$feet is $0.0808$.

c. The ${80}^{th}$percentile of the distribution of the average of $49$fly balls is$246$

Step-by-step solution

Given information part (a)

Given in the question that, the distance of fly balls hit to the outfield is normally distributed with a mean of $250$feet and a standard deviation of $50$feet. We randomly sample $49$fly balls.

Explanation (Part a)

Consider $\overline{X}$ as the average distance in feet for $49$fly balls, then

localid="1649344986876" $\overline{X}~N\left(250,\frac{50}{\sqrt{49}}\right)$

Given Information (part b)

According to the information,

$\overline{X}~N\left(250,\frac{50}{\sqrt{49}}\right)=N(250,7.143)$

Explanation (Part b)

Let's find the probability:

$P(\overline{X}<240)=P\left(\frac{\overline{X}-250}{7.143}<\frac{240-250}{7.143}\right)$

$=P\left(\frac{\overline{x}-250}{7.143}<-1.4\right)$

$=\varphi (-1.4)$

$=1-0.9192$

$=0.0808$

Graphical Representation (Part b)

Given Information (Part c)

The distance of fly balls hit to the outfield is normally distributed with a mean of $250$feet and a standard deviation of $50$feet. We randomly sample $49$fly balls.

Explanation (Part c)

Let's find the value of $z$from

$P(\mathrm{\Sigma }x<z)=0.8$

$P\left(\frac{\mathrm{\Sigma }x-240}{7.143}<\frac{z-240}{7.143}\right)=0.8$

If a variable $z$has a normal distribution, then

$P(Z<0.84)=0.8$

Now we have

$\frac{z-240}{7.143}=0.84$

$z-240=6$

$z=246$