The linear regression equation that represents this data set is equal to y = 120.8x + 2,114.
How to write the linear regression equation?
First of all, we would determine the slope of the given data set by using this formula:
[tex]Slope = \frac{\sum (x-\bar x)(y-\bar y)}{\sum (x-\bar x)^2}[/tex]
For the sample mean (years), we have:
[tex]\bar x = \frac{0\;+\;1+\;2\;+\;3\;+4}{5} \\\\\bar x =\frac{15}{4} \\\\\bar x =3.75[/tex]
For the sample mean (cases), we have:
[tex]\bar y = \frac{904\;+\;900+\;833\;+\;837\;+754}{5} \\\\\bar y =\frac{4,228}{4} \\\\\bar y =1057[/tex]
The sums are given by:
∑(x - |x|) = (0 - 3.75) + (1 - 3.75) + (2 - 3.75) + (3 - 3.75) + (4 - 3.75)
∑(x - |x|) = - 3.75 - 2.75 - 1.75 - 0.75) + 0.25
∑(x - |x|) = -8.75.
∑(x - |x|)² = (-8.75)² = 76.5625.
∑(y - |y|) = (904 - 1057) + (900 - 1057) + (833 - 1057) + (837 - 1057) + (754 - 1057)
∑(y - |y|) = -153 - 157 - 224 - 220 - 303
∑(y - |y|) = -1,057.
∑(x - |x|)(y - |y|) = -8.75 × -1057
∑(x - |x|)(y - |y|) = 9,248.75.
Now, we can determine the slope:
Slope = 9,248.75/76.5625
Slope = 120.8.
Therefore, the equation is given by:
y = 120.8x + a
Next, we would find the value of a:
1057 = 120.8(-8.75) + a
a = 1057 + 1,057
a = 2,114.
Hence, the linear regression model is given by:
y = 120.8x + 2,114.
Since the new cases would reach 579, the calendar year would be calculated as follows:
579 = 120.8x + 2,114
120.8x = 579 - 2,114
120.8x = -1,535
x = -1,535/120.8
x = -12.71.
Year = 2014 + 12.71
Year = 2026.71 ≈ 2027.
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