As online attacks persist, spending on information security software continues to rise. The table gives the forecast for the worldwide sales (in billions of dollars) of information security software through 2007 (t = 0 corresponds to 2002):
Year, t 0 1 2 3 4 5
Spending, y 6 8.9 11.9 14 15.7 16.2
i put y=2.1x+6.87 as the least squares equation but it's wrong. HELP PLEASE!
use x for t
y = a + b x
x mean = 15/6 = 2.5
y mean = 72.7/6 = 12.12
x-x mean
-2.5 -1.5 -.5 +.5 +1.5 + 2.5
y - y mean
-6.12 -3.22 -.22 +1.88 +3.58 +4.08
(x - x mean)(y - y mean)
15.3 4.83 .11 .94 5.37 10.2
sum of those = 36.75
(x - x mean)^2
6.25 2.25 .25 .25 2.25 6.25
sum of those = 17.5
so
b = 36.75/17.5 = 2.1 agree with you
a = y mean - b x mean
= 12.12 - 2.1* 2.5 = 6.87 agree with you
y = 6.87 + 2.1 t
perhaps you used x in last line instead of t?
or
perhaps they want real years and billions
y = 10^9 (6.87 + 2.1(year-2002) )
i used x instead of t. thanks
To find the least squares equation, we need to perform linear regression on the given data. The equation for linear regression has the form y = mx + b, where y represents the sales (dependent variable), x represents the year (independent variable), m represents the slope, and b represents the y-intercept.
Let's calculate the slope (m) and y-intercept (b) step by step:
Step 1: Calculate the mean of the independent variable (x) and dependent variable (y):
- Mean of x = (0 + 1 + 2 + 3 + 4 + 5) / 6 = 15 / 6 = 2.5
- Mean of y = (6 + 8.9 + 11.9 + 14 + 15.7 + 16.2) / 6 ≈ 72.7 / 6 ≈ 12.12
Step 2: Calculate the sum of the products of the differences between each x value and the mean of x (x - mean of x) and the corresponding differences between each y value and the mean of y (y - mean of y):
- Sum of [ (x - mean of x) * (y - mean of y) ] = (0 - 2.5) * (6 - 12.12) + (1 - 2.5) * (8.9 - 12.12) + (2 - 2.5) * (11.9 - 12.12) + (3 - 2.5) * (14 - 12.12) + (4 - 2.5) * (15.7 - 12.12) + (5 - 2.5) * (16.2 - 12.12) = (-2.5) * (-6.12) + (-1.5) * (-3.22) + (-0.5) * (-0.22) + (0.5) * (1.88) + (1.5) * (3.58) + (2.5) * (4.08) ≈ 15.3 + 4.83 + 0.11 + 0.94 + 5.37 + 10.2 ≈ 36.77
Step 3: Calculate the sum of the squares of the differences between each x value and the mean of x [(x - mean of x)^2]:
- Sum of [ (x - mean of x)^2 ] = (0 - 2.5)^2 + (1 - 2.5)^2 + (2 - 2.5)^2 + (3 - 2.5)^2 + (4 - 2.5)^2 + (5 - 2.5)^2 = ( -2.5)^2 + (-1.5)^2 + (-0.5)^2 + (0.5)^2 + (1.5)^2 + (2.5)^2 = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5
Step 4: Calculate the slope (m):
- m = [ Sum of [ (x - mean of x) * (y - mean of y) ] ] / [ Sum of [ (x - mean of x)^2 ] ] = 36.77 / 17.5 ≈ 2.103
Step 5: Calculate the y-intercept (b):
- b = mean of y - (m * mean of x) = 12.12 - (2.103 * 2.5) ≈ 6.864
Therefore, the least squares equation for the given data is y ≈ 2.103x + 6.864.
Note: It seems like you made a minor rounding error when calculating the slope (m) and y-intercept (b), which resulted in a slightly different equation.