The Normal Distribution - Homework

Question

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 254 feet and a standard deviation of 58 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 216 feet?

1. P�(fewer than 216 feet) = %

b) If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled more than 240 feet?

1. P�(more than 240 feet) = %

Answer 1

Since the directions state that you can use your graphing calculator it is possible that you will not have to do much work for this question. I'm going to present two possible options here since both require the use of your calculator.

Method 1: Converting to a z-score and then using the standard normal model in your calculator.

For any variable that is normally distributed with a mean of µ and a standard deviation of σ we can convert that normal model to what is called the standard normal model (which is the same bell-shaped curve but it specifically has a mean of 0 and a standard deviation of 1). Converting to this model means that not only would you be able to then use your calculator to find the area under the curve but you would also be able to use a standard normal table. Finding the z-score is also useful because the z-score tells us how many standard deviations above/below the mean a value lies.

Suppose the question had asked for P(x < 230) aka fewer than 230 feet.

First, we would convert to a z-score by using the formula:

z = (x - µ)/σ

For this specific example it would be:

z = (230 - 254)/58 = -0.4138

This means that a fly ball that travels 230 feet is traveling .4138 standard deviations less than the average distance. However, we are looking for the probability that the ball would travel FEWER than 230 feet. To find this probability, we would use normalcdf in our calculator (To find this in a TI-83 or 84, click 2nd-->VARS-->normalcdf) and input -999999 as the lower bound, -0.4138 as the upper bound, µ = 0, and σ = 1 so normalcdf(-999999, -0.4138, 0, 1). The reason we keep 0 and 1 as the mean and standard deviation is because converting to a z-score means we converted to the model with that mean and standard deviation.

As an answer we get normalcdf(-999999, -0.4138, 0, 1) = .3395

So this means the probability of a ball traveling fewer than 230 feet would be 33.95%.

If instead we had been asked for the probability that the ball would travel more than 230 feet, we still would have found the z-score but then instead of looking for the area between -999999 and -0.4138, we would have looked for the area above -0.4138, so we would have input normalcdf(-0.4138, 999999, 0, 1).

Method 2: Use the calculator to do ALL of the work (No z-score conversion needed)

Your calculator can also jump directly to calculating the probability of being greater than, less than, or between two values on the normal curve without us having to convert the original value to a z-score.

Using the same example as before: P(X < 230), we could simply use normalcdf as follows:

normalcdf(-999999, 230, 254, 58) where we enter the actual value of 230 instead of the z-score, and 254 as µ and 58 as σ since we did not convert to the standard normal and are simply using the Normal model described in the model. Doing this gets us the same answer of 33.95%

Hopefully you can now try to follow that process to answer the questions P(X < 216) and P(X >240)