Question

1.) 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. Using Excel and the functions show me how you got your answer

1. If¯¯¯¯¯XX¯ = average distance in feet for 49 fly balls, then¯¯¯¯¯XX¯ ~ _______(_______,_______)
2. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for¯¯¯¯¯XX¯. Shade the region corresponding to the probability. Find the probability. using excel functions
3. Find the 80^th percentile of the distribution of the average of 49 fly balls. using excel functions

2.) Suppose that a category of world-class runners are known to run a marathon (26 miles) in an average of 145 minutes with a standard deviation of 14 minutes. Consider 49 of the races. Let¯¯¯¯¯XX¯ the average of the 49 races. Using Excel and the excel functions for the following questions. Show me how you got your answer

1. ¯¯¯¯¯XX¯ ~ _____(_____,_____)
2. Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons. using excel functions
3. Find the 80^th percentile for the average of these 49 marathons. using excel functions
4. Find the median of the average running times. using excel functions

Answer 1

1. Let X be the distance in feet of any randomly chosen baseball. Given
Let be the average distance in feet for the 49 balls, then using the Central Limit Theorem

using

The probability that the 49 balls traveled an average of less than 240 feet =

The below functions shows how to calculate the above probability using Excel.

•    using NORMDIST(x=-1.4, mean=0, std=1, cumulative=TRUE) = 0.08

OR

• using NORMDIST(x=240, mean=250, std=7.14, cumulative=TRUE) = 0.08

See the below plot for the shaded area

To randomly sample 49 observations in Excel use NORMINV(RAND(), mean=250, standard_dev=50) and drag the cell downwards to 49 rows. This will generate 49 random values from the given normal distribution

To find the 80th percentile of the average of the distance of 49 baseballs using Excel use
NORMINV(probability=0.8, mean=250, standard_dev=7.14) = 256.0092

2. Let X be the running time of any randomly chosen worldclass runnder. Given

Let be the average running time for the 49 races, then using the Central Limit Theorem

using

The  probability that the runner will average between 142 and 146 minutes in these 49 marathons =

This can be calculated in Excel using NORMDIST(146, 145, 2, TRUE) - NORMDIST(142, 145, 2, TRUE) = 0.6246

To find the 80th percentile of the distribution using Excel use
NORMINV(probability=0.8, mean=145, standard_dev=2) = 146.6832

To find the median of the 49 races using Excel use
NORMINV(probability=0.5, mean=145, standard_dev=2) = 145