Given are five observations for two variables x and y Round

Given are five observations for two variables, x and y.

Round your answers to two decimal places.

Estimate the standard deviation of * when x = 3.

Develop a 95% confidence interval for the expected value of y when x = 3.

Estimate the standard deviation of an individual value of y when x = 3.

Develop a 95% prediction interval for y when x = 3.

xi	1	2	3	4	5
yi	4	7	6	12	13

Solution

Given are five observations for two variables, x and y.

We can compute all these using EXCEL.

1) Standard deviation of y^* using the equation

Sy^* = s * sqrt [ 1/n + (x* - xbar)^2 / (xi - xbar)^2 ]

where s = sqrt [ ( SSy - b*SSxy) / (n-2) ]

SSy = y² - (y)² / n

SSxy = xy - (x)*(y) / n

n = 5

The table for all these values as below:

First we find the regression equation

y^ = a + b*x

where a is intercept

b is slope

EXCEL syntax for a and b is,

=intercept (known y\'s ,known x\'s)

= slope (known y\'s ,known x\'s)

a = 1.5 and b = 2.3

So the regression equation is,

y^ = 1.5 + 2.3 * x

Now we find from the table SSy and SSxy using table,

SSy = 414 - (42^2/5) = 61.2

SSxy = 149 - (15*42)/5 = 23

s = sqrt [ (61.2 - 2.3*23) / 5-2 ] = 1.66

given that at x = 3 we have to calculate standard deviation os y^*.

when x* = 3

xbar = sum of x values / total number of values = 15

(x* - xbar)² = 0

Sy^* = s * sqrt [ 1/n + [ (x* - xbar)² / (xi - xbar)² ]

Sy^* = 1.66 * sqrt [ 1/5 + [0 / 10 ]

Sy^* = 0.74

Now using the following equation find 95% confodence interval for the expected value of y when x = 3.

y^ * + - t_alpha/2 * Sy^*

where alpha = 0.05

t_alpha/2 =

EXCEL syntax :

=tinv(probability,d.f.)

where probability = alpha

d.f. = n - 2 = 5 - 2 = 3

t_alpha/2 = 3.18

y^* when x = 3

y^* = 1.5 + (3 * 2.3) = 8.4

lower limit = y^* -  t_alpha/2 * Sy^* = 8.4 - 3.18 * 0.74 = 6.05

upper limit = y^* + t_alpha/2 * Sy^* = 8.4 + 3.18 * 0.74 = 10.75

confidence interval for expected value of y is (6.05 , 10.75).

Now we have to find Spred.

Spred = s * sqrt [ 1 + 1/n + (x*-xbar)² / (xi-xbar)²  ]

= 1.66 * sqrt [ 1 + 1/5 + (0/10)² ]

= 1.82

Now we have to calculate 95% conficence interval for y when x = 3

lower limit = y^* -  t_alpha/2 *Spred = 8.4 - (3.18*1.82) = 2.61

lower limit = y^* + t_alpha/2 *Spred = 8.4 + (3.18*1.82) = 14.19

confidence interval for y when x = 3 is (2.61 , 14.19)

xi	yi	xi^2	yi^2	xiyi	(xi-xbar)^2
1	4	1	16	4	4
2	7	4	49	14	1
3	6	9	36	18	0
4	12	16	144	48	1
5	13	25	169	65	4
15	42	55	414	149	10