Standard deviation/probability

Question

Peter and Sally each shoot 12 arrows at a target that is circular with a diameter of 72 inches. the mean of the distances that Peter's arrows strike from the center of the target is 20 inches, and the standard deviation of his distance is 3 inches. The mean of the distances that Sally's arrows strike from the center of the target is 16 inches, and the standard deviation of her distances is 4 inches. What statements can be made about the region of the target that at least 75 percent of Peter's arrows strike and the region of the target that at least 75 percent of Sally's arrows strike?

Answer 1

It sounds like we want to find the regions of the target (defined by the radius or diameter) where 75% of the arrows strike.

In order to do that, we want to use a z-table along with the formula, z = (x - μ ) / σ. However, instead of solving for z and then looking it up in the z-table to obtain a percentage, we are going to go in the opposite direction. Since we want to know 75% or more, let's look in the z-table for when it hits .75. At the intersection of 0.6 and .07, which means a z-score of 0.67, we have the value .7486, which is 74.68%, and at the intersection of 0.6 and .08, we have a z-score of 0.68 pertaining to the value .7517, or 75.17%. Let's go with the latter, z = 0.68.

So cool, now we have the z-value! What we want now is to use the z value, along with the given information to solve for x in the above equation, as x is the amount of inches from the center of the target that pertain to 75% of the 12 shots.

z = (x - μ ) / σ

keeping the same relationship, we can do some algebra to solve for x:

x = (z * σ) + μ

Peter:

x = (z * σ) + μ = (.68 * 3) + 20 = 22.04

this means that at least 75% of Peter's arrows ended up 22.04 inches from the center of the target

Sally:

x = (z * σ) + μ = (.68 * 4) + 16 = 18.72

this means that at least 75% of Sally's arrows ended up 18.72 inches from the center of the target

You could use these radii to draw conclusions about the size (area) of the regions if you wish to go further.

Best,

Carly