The following table shows the number of wins e ght teams had during a football s

ID: 3351598 • Letter: T

Question

The following table shows the number of wins e ght teams had during a football season. Also shown are the average ponts each team scored per game du teams that scored an average of 26 points a game the season onstruct a 90% pre ction rite va t estimate the number o n s for Click the icon to view the data table. Determine the upper and lower limits of the prediction interval, UPL- LPL = Round to two decimal places as needed.) Data Table Wins 14 8 3 9 2 6 13 8 25.9 18.8 20.4 24.4 12.3 22.5 22.9 23.1 Points per Print Done

Explanation / Answer

Back-up Theory

Let X = average points per game and Y = number of wins during the season.

We want to get a 90% Prediction Interval (PI) to estimate number of wins when the average points per game is 26.

i.e., we want an interval estimate for y corresponding to x = 26.

The linear regression model assumed is: Y = + X + , where is the error term, which is assumed to be Normally distributed with mean 0 and variance 2.

Let (xi, yi) be a pair of sample observation on (X, Y), i= 1, 2, …., n, where n = sample size.

Mean X = Xbar = (1/n)sum of xi over I = 1, 2, …., n; ……………….(1)

Sxx = sum of (xi – Xbar)2 over i = 1, 2, …., n ………………………………..(2)

Similarly, Mean Y = Ybar =(1/n)sum of yi over i= 1, 2, …., n;…………….(3)

Syy = sum of (yi – Ybar)2 over i = 1, 2, …., n ………………………………………………(4)

Sxy = sum of {(xi – Xbar)(yi – Ybar)} over i = 1, 2, …., n………(5)

Estimated Regression of Y on X is given by: Ycap = a + bX, where

b = Sxy/Sxx ……………………………………………………………………….(6) and

a = Ybar – bYX.Xbar……………………………………..…………………….(7)

Estimate of 2 is given by s2 = (Syy – b2Sxx)/(n - 2)…………………………………(8)

100(1 - )% Prediction Interval (PI) for ycap at x = x0 is (a + bx0) ± tn – 2, /2xs[1 + (1/n) + {(x0 – Xbar)2/Sxx}] ………………………………………(9)

Calculations

25.9

18.8

20.4

24.4

12.3

22.5

22.9

23.1

xbar

ybar

7.875

Sxx

126.06875

Syy

126.875

Sxy

94.7875

0.7519

-8.1305

PI LB

2.921

PIUB

19.915