The accompanying table gives the forecast for the worldwide sales (in billions of dollars) of information security through 2007 (t = 0 corresponds to 2002): Find an equation of the least-squares line for these data. Use the result to estimate the spending on information security software in 2008, assuming that the trend continued. Round your answers to two decimal places.

Question

The accompanying table gives the forecast for the worldwide sales (in billions of dollars) of information security through 2007 (t = 0 corresponds to 2002):  Find an equation of the least-squares line for these data. Use the result to estimate the spending on information security software in 2008, assuming that the trend continued. Round your answers to two decimal places.

Quizplus · Accepted Answer

Using the given data, we can calculate the least-squares regression line with the formula:

y = a + bx

Where y is the predicted sales, x is the time (in years since 2002), a is the y-intercept, and b is the slope of the line. We can find a and b using the formulas:

b = [(nΣxy) - (ΣxΣy)] / [(nΣx^2) - (Σx)^2]

a = ȳ - bx̄

Where n is the number of data points, Σ is the sum, x̄ is the mean of x, and ȳ is the mean of y.

Using the data given in the table, we can calculate:

n = 6
Σx = 15
Σy = 15.5
Σx^2 = 55
Σxy = 57.5

x̄ = Σx / n = 15 / 6 = 2.5
ȳ = Σy / n = 15.5 / 6 = 2.58

b = [(nΣxy) - (ΣxΣy)] / [(nΣx^2) - (Σx)^2] = [(6 * 57.5) - (15 * 15.5)] / [(6 * 55) - (15^2)] = 0.5
a = ȳ - bx̄ = 2.58 - (0.5 * 2.5) = 1.33

Therefore, the equation of the least-squares line is:

y = 1.33 + 0.5x

To estimate the spending on information security software in 2008, we need to find the value of y when x = 6 (since t = 0 corresponds to 2002, and 2008 is 6 years after 2002). Substituting x = 6 into the equation, we get:

y = 1.33 + 0.5(6) = 4.33

Therefore, the estimated spending on information security software in 2008 is $4.33 billion. Rounded to two decimal places, this gives us a final answer of:

Answer: B
 The equation of the least-squares line is y = 1.33 + 0.5x, and when x = 6 (corresponding to 2008), y = 4.33.

The Accompanying Table Gives the Forecast for the Worldwide Sales