SOLUTION: The following example describes the expenditure (in dollars) on recreation per month by employees at a certain company, and their corresponding monthly incomes. We want to use recr

Question 1188606: The following example describes the expenditure (in dollars) on recreation per month by employees at a certain company, and their corresponding monthly incomes. We want to use recreational expenditures to predict monthly incomes. (30 points total)
Expenditure ($) 2400, 2650 ,2350 ,4950 ,3100 ,2500 , 5106 , 3100 , 2900 , 1750
Income ($) 41200, 50100, 52000, 66000, 44500, 37700, 73500, 37500, 56700, 35600
(a) Calculate the linear regression line for the data. (5 points)
(b) What is the r value? Is there a positive or negative correlation? (5 points)
(c) Give a practical interpretation of the slope and the intercept in the context of this problem, respectively. Be sure to write complete and grammatically correct sentences. (10 points)
(d) Using the equation for the linear regression that you calculated, estimate the monthly income of an employee at this company who spends 5000 dollars per month on recreation. Give a practical interpretation of this prediction in the context of this problem. (10 points)

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part (a)

You could calculate the regression line by hand, but it's tedious busywork in my opinion. Instead, it's better to rely on technology. There are many free options out there that will calculate things like regression lines, correlation coefficients, etc. Many if not all spreadsheet programs can do this, not to mention specialized websites as well.

After using a spreadsheet program, the regression line I get is
y = 9.77x + 19370.11
Both decimal values are approximate.

x = monthly expenditure on recreation
y = monthly income
both are in dollars

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Part (b)

Using the same spreadsheet program, I get this approximate r value
r = 0.8404

Like with part (a), it's possible to calculate by hand through a formula, but it's preferable to use technology. This is positive correlation because the r value is positive. The regression line has a positive slope which is more evidence we have positive correlation. As x goes up, so does y.
Since r is fairly close to 1, this tells the reader that the linear correlation is fairly strong.

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Part (c)

Earlier in part (a), we found this regression line
y = 9.77x + 19370.11

The slope of 9.77 tells us that each time x goes up by 1, y roughly increases by 9.77. To put it in more specific context, it means that each time the expenditure x goes up by $1, the monthly income y goes up by about $9.77

The y intercept of 19370.11 indicates that when the expenditure is x = 0, then the monthly income is predicted to be about $19370.11; we can think of this as the initial monthly income so to speak. This is because we can't go any lower than x = 0. Negative x values aren't allowed.

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Part (d)

Plug x = 5000 into the regression equation and simplify

y = 9.77x + 19370.11
y = 9.77*5000 + 19370.11
y = 48850 + 19370.11
y = 68220.11

We estimate that someone spending $5000 per month on recreation will have a monthly income of about $68,220.11

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