Answer:
(a) y = -.613x +4.142
Explanation:
You want the equation of the line of best fit for the given data.
Line of best fit
The line of best fit will minimize the sum of the squares of the residuals. Those are the differences between the actual data value and the data predicted by the line.
The slope of the line of best fit is the ratio of the covariance of x and y to the variance of x. The line goes through the point (µx, µy). There are several ways you can arrive at the equation for the line of best fit:
- use the linear regression function of a spreadsheet or calculator
- find the slope and intercept from means and variances
- use formulas for the slope and intercept
- "eyeball" a reasonable line through the plotted data.
Calculator
Many graphing calculators, and all spreadsheets, have functions for computing a line of best fit for a data set. The first attachment shows the linear regression function results for a TI-84 work-alike calculator app. It shows you the equation of the line is approximately ...
y = ax +b = -0.613x +4.142 . . . . . . matches the first choice
The second attachment shows the same result achieved using the correlation and mean functions provided by the calculator. Again, it tells you the equation is ...
y = -0.613x +4.142
Choices
If you examine the given data, you see that generally as x gets larger, y gets smaller. This tells you the line has a negative slope, eliminating choices C and D.
The point (x, y) = (0, 4) tells you the y-intercept is positive, and near +4. This eliminates choice B.
The only reasonable choice among those offered is choice A.
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Additional comment
We have not shown the formulas for the parameters of the line of best fit. The usual formulas for variance, covariance, and mean apply. There are a number of different ways these can be calculated, some easier than others, depending on the calculator being used. For a multiple-choice problem like this, the analysis of the choices offers the quickest solution.
The third attachment shows a plot of the data, along with the line of best fit.
(When the offered choices differ from each other by only a few percent, then calculation is needed for choosing the correct one.)
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