Answer :
(a) To find the relevant sums based on the given data:
We have:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 95 \\ 10 & 81 \\ 18 & 70 \\ 22 & 50 \\ 28 & 73 \\ 20 & 94 \\ 12 & 90 \\ 15 & 62 \\ \hline \end{array} \][/tex]
Note: The value for [tex]\(y\)[/tex] when [tex]\(x=25\)[/tex] is missing.
First, let’s sum up the given [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values.
[tex]\[ \sum x = 0 + 10 + 18 + 22 + 28 + 20 + 12 + 15 = 125 \][/tex]
[tex]\[ \sum y = 95 + 81 + 70 + 50 + 73 + 94 + 90 + 62 = 615 \][/tex]
Next, we calculate the sum of the squares of [tex]\(y\)[/tex] values:
[tex]\[ \sum y^2 = 95^2 + 81^2 + 70^2 + 50^2 + 73^2 + 94^2 + 90^2 + 62^2 = 9025 + 6561 + 4900 + 2500 + 5329 + 8836 + 8100 + 3844 = 49095 \][/tex]
Now, we calculate the sum of the products of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
[tex]\[ \sum xy = (0 \cdot 95) + (10 \cdot 81) + (18 \cdot 70) + (22 \cdot 50) + (28 \cdot 73) + (20 \cdot 94) + (12 \cdot 90) + (15 \cdot 62) = 0 + 810 + 1260 + 1100 + 2044 + 1880 + 1080 + 930 = 9104 \][/tex]
Lastly, we calculate the sum of the squares of [tex]\(x\)[/tex] values:
[tex]\[ \sum x^2 = 0^2 + 10^2 + 18^2 + 22^2 + 28^2 + 20^2 + 12^2 + 15^2 = 0 + 100 + 324 + 484 + 784 + 400 + 144 + 225 = 2461 \][/tex]
Thus the sums are:
[tex]\[ \sum x = 125, \quad \sum y = 615, \quad \sum y^2 = 49095, \quad \sum xy = 9104, \quad \sum x^2 = 2461 \][/tex]
(b) To find the correlation coefficient (r):
Using the formula:
[tex]\[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{n \sum x^2 - (\sum x)^2} \sqrt{n \sum y^2 - (\sum y)^2}} \][/tex]
Where [tex]\(n = 8\)[/tex] (excluding the entry where [tex]\(y\)[/tex] is missing):
[tex]\[ r = \frac{8 \times 9104 - 125 \times 615}{\sqrt{8 \times 2461 - 125^2} \sqrt{8 \times 49095 - 615^2}} = \frac{72832 - 76875}{\sqrt{19688 - 15625} \sqrt{392760 - 378225}} = \frac{-4043}{\sqrt{4063} \sqrt{14535}} = \frac{-4043}{\sqrt{59079855}} = -0.526 \][/tex]
(c) To find the standard deviations of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
Standard deviation formula for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ s_x = \sqrt{\frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n-1}} = \sqrt{\frac{2461 - \frac{125^2}{8}}{7}} = \sqrt{\frac{2461 - 1953.125}{7}} = \sqrt{\frac{507.875}{7}} = 8.518 \][/tex]
[tex]\[ s_y = \sqrt{\frac{\sum y^2 - \frac{(\sum y)^2}{n}}{n-1}} = \sqrt{\frac{49095 - \frac{615^2}{8}}{7}} = \sqrt{\frac{49095 - 47265.625}{7}} = \sqrt{\frac{1829.375}{7}} = 16.111 \][/tex]
(d) To find the slope [tex]\(a\)[/tex] of the best-fit line:
Using the formula:
[tex]\[ a = r \frac{s_y}{s_x} = -0.526 \times \frac{16.111}{8.518} = -0.995 \][/tex]
(e) To find the y-intercept [tex]\(b\)[/tex] of the best-fit line:
Using the formula:
[tex]\[ b = \frac{\sum y}{n} - a \frac{\sum x}{n} = \frac{615}{8} - (-0.995) \times \frac{125}{8} = 76.875 + 15.548 = 92.423 \][/tex]
Therefore, the values are filled in correctly as follows:
- [tex]\(\sum x = 125\)[/tex]
- [tex]\(\sum y = 615\)[/tex]
- [tex]\(\sum y^2 = 49095\)[/tex]
- [tex]\(\sum xy = 9104\)[/tex]
- [tex]\(\sum x^2 = 2461\)[/tex]
- The correlation coefficient [tex]\( r = -0.526\)[/tex]
- The standard deviations [tex]\( s_x = 8.518\)[/tex] and [tex]\( s_y = 16.111\)[/tex]
- The slope of the best fit line [tex]\(a = -0.995\)[/tex]
- The y-intercept of the best fit line is [tex]\(b = 92.423\)[/tex]
We have:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 95 \\ 10 & 81 \\ 18 & 70 \\ 22 & 50 \\ 28 & 73 \\ 20 & 94 \\ 12 & 90 \\ 15 & 62 \\ \hline \end{array} \][/tex]
Note: The value for [tex]\(y\)[/tex] when [tex]\(x=25\)[/tex] is missing.
First, let’s sum up the given [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values.
[tex]\[ \sum x = 0 + 10 + 18 + 22 + 28 + 20 + 12 + 15 = 125 \][/tex]
[tex]\[ \sum y = 95 + 81 + 70 + 50 + 73 + 94 + 90 + 62 = 615 \][/tex]
Next, we calculate the sum of the squares of [tex]\(y\)[/tex] values:
[tex]\[ \sum y^2 = 95^2 + 81^2 + 70^2 + 50^2 + 73^2 + 94^2 + 90^2 + 62^2 = 9025 + 6561 + 4900 + 2500 + 5329 + 8836 + 8100 + 3844 = 49095 \][/tex]
Now, we calculate the sum of the products of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
[tex]\[ \sum xy = (0 \cdot 95) + (10 \cdot 81) + (18 \cdot 70) + (22 \cdot 50) + (28 \cdot 73) + (20 \cdot 94) + (12 \cdot 90) + (15 \cdot 62) = 0 + 810 + 1260 + 1100 + 2044 + 1880 + 1080 + 930 = 9104 \][/tex]
Lastly, we calculate the sum of the squares of [tex]\(x\)[/tex] values:
[tex]\[ \sum x^2 = 0^2 + 10^2 + 18^2 + 22^2 + 28^2 + 20^2 + 12^2 + 15^2 = 0 + 100 + 324 + 484 + 784 + 400 + 144 + 225 = 2461 \][/tex]
Thus the sums are:
[tex]\[ \sum x = 125, \quad \sum y = 615, \quad \sum y^2 = 49095, \quad \sum xy = 9104, \quad \sum x^2 = 2461 \][/tex]
(b) To find the correlation coefficient (r):
Using the formula:
[tex]\[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{n \sum x^2 - (\sum x)^2} \sqrt{n \sum y^2 - (\sum y)^2}} \][/tex]
Where [tex]\(n = 8\)[/tex] (excluding the entry where [tex]\(y\)[/tex] is missing):
[tex]\[ r = \frac{8 \times 9104 - 125 \times 615}{\sqrt{8 \times 2461 - 125^2} \sqrt{8 \times 49095 - 615^2}} = \frac{72832 - 76875}{\sqrt{19688 - 15625} \sqrt{392760 - 378225}} = \frac{-4043}{\sqrt{4063} \sqrt{14535}} = \frac{-4043}{\sqrt{59079855}} = -0.526 \][/tex]
(c) To find the standard deviations of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
Standard deviation formula for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ s_x = \sqrt{\frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n-1}} = \sqrt{\frac{2461 - \frac{125^2}{8}}{7}} = \sqrt{\frac{2461 - 1953.125}{7}} = \sqrt{\frac{507.875}{7}} = 8.518 \][/tex]
[tex]\[ s_y = \sqrt{\frac{\sum y^2 - \frac{(\sum y)^2}{n}}{n-1}} = \sqrt{\frac{49095 - \frac{615^2}{8}}{7}} = \sqrt{\frac{49095 - 47265.625}{7}} = \sqrt{\frac{1829.375}{7}} = 16.111 \][/tex]
(d) To find the slope [tex]\(a\)[/tex] of the best-fit line:
Using the formula:
[tex]\[ a = r \frac{s_y}{s_x} = -0.526 \times \frac{16.111}{8.518} = -0.995 \][/tex]
(e) To find the y-intercept [tex]\(b\)[/tex] of the best-fit line:
Using the formula:
[tex]\[ b = \frac{\sum y}{n} - a \frac{\sum x}{n} = \frac{615}{8} - (-0.995) \times \frac{125}{8} = 76.875 + 15.548 = 92.423 \][/tex]
Therefore, the values are filled in correctly as follows:
- [tex]\(\sum x = 125\)[/tex]
- [tex]\(\sum y = 615\)[/tex]
- [tex]\(\sum y^2 = 49095\)[/tex]
- [tex]\(\sum xy = 9104\)[/tex]
- [tex]\(\sum x^2 = 2461\)[/tex]
- The correlation coefficient [tex]\( r = -0.526\)[/tex]
- The standard deviations [tex]\( s_x = 8.518\)[/tex] and [tex]\( s_y = 16.111\)[/tex]
- The slope of the best fit line [tex]\(a = -0.995\)[/tex]
- The y-intercept of the best fit line is [tex]\(b = 92.423\)[/tex]
