(2) For the sport of NFL football during the 2014 season there were 1293 touchdo

ID: 3207574 • Letter: #

Question

(2) For the sport of NFL football during the 2014 season there were 1293 touchdowns scored during the regular season. There were 256 games played during the season. You will use the Poisson distribution to estimate probabilities of games with a certain number of touchdowns. (a, 4 pts) What was the mean number of touchdowns TD) scored in each mean TD game during the season? (round the answer to the nearest 0.0001) (b, 1 pt ea) Complete the chart at the right. EATD Probability Predicted games The first column lists the number of touchdowns in a game, this is filled in already. The second column is for the predicted probability that 1 a game chosen at random will have that many touchdowns scored, calculate these values, round these values to the closest 0.0001 The third column is for your best prediction about the number of games during the season that had that many 4 touchdowns scored, round these values to the closest whole number. 10 (c, 4 pts) on Jan 10, 2010 The Green Bay Packers and Arizona Cardinals played a playoff game in which there were 13 touchdowns scored. Using the data above, what is the probability that a random game would have that many Prob (13+TD) or more touchdowns? (round the answer to the nearest 0.0001) G 45 51 Arizona Packers 1 2 3 4 Cardinals

Explanation / Answer

Solution

Back-up Theory

If X ~ Poisson with parameter , then P(X = x) = e-()x/(x!) where = average, and

P(X x) = Sum over x = 0 to x of {e-()x/(x!)} and

P(X > x) = 1 - Sum over x = 0 to x of {e-()x/(x!)}

Now, to work out the solution,

Given 1293 touchdowns in 256 games, = mean number of touchdowns per game = 1293/256

= 5.05078 = 5.0508 (rounded to 4 places)

Let X = number of touchdowns in a game. Then, X ~ Poisson (5.0508)

Part (a)(1)

As already obtained above, mean number of touchdowns per game = 1293/256

= 5.05078 = 5.0508 (rounded to 4 places) ANSWER

Part (b)(2)

The required probability and predicted # of games

are given in the following table.

Predicted probability given in column 2 is

obtained using Excel Function.

Predicted # games is obtained by multiplying

p(x) in column 2 by 256.

p(x)

Predicted # games

0.0064

0.0323

0.0817

0.1375

0.1737

0.1754

0.1477

0.1066

0.0673

0.0378

0.0191

Part (c)(3)

We want P(X 13) = 1 – P(X < 13) = 1 – 0.9992 = 0.0008 ANSWER