Question

what is the approximate correlation coefficient of the linear best - fit model for the data set shown below?

x	y
10	80
7	90
20	220
30	440
40	520
50	630
60	730
70	900
35	400
55	820
45	790

\\(\bigcirc\\) 0.9468
\\(\bigcirc\\) 0.9577
\\(\bigcirc\\) 0.9083
\\(\bigcirc\\) 0.9707

Explanation:

Step1: Recall the formula for correlation coefficient \( r \)

The formula for the Pearson correlation coefficient \( r \) is:
\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]
where \( n \) is the number of data points.

Step2: Calculate necessary sums

First, list the data points \((x, y)\):
\((5, 25)\), \((10, 80)\), \((7, 90)\), \((20, 220)\), \((30, 440)\), \((40, 520)\), \((50, 630)\), \((60, 730)\), \((70, 900)\), \((35, 400)\), \((55, 820)\), \((45, 790)\)
So \( n = 12 \).

Calculate \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), \( \sum y^2 \):

- \( \sum x = 5 + 10 + 7 + 20 + 30 + 40 + 50 + 60 + 70 + 35 + 55 + 45 \)
\( = 5 + 10 = 15 \); \( 15 + 7 = 22 \); \( 22 + 20 = 42 \); \( 42 + 30 = 72 \); \( 72 + 40 = 112 \); \( 112 + 50 = 162 \); \( 162 + 60 = 222 \); \( 222 + 70 = 292 \); \( 292 + 35 = 327 \); \( 327 + 55 = 382 \); \( 382 + 45 = 427 \)
- \( \sum y = 25 + 80 + 90 + 220 + 440 + 520 + 630 + 730 + 900 + 400 + 820 + 790 \)
\( 25 + 80 = 105 \); \( 105 + 90 = 195 \); \( 195 + 220 = 415 \); \( 415 + 440 = 855 \); \( 855 + 520 = 1375 \); \( 1375 + 630 = 2005 \); \( 2005 + 730 = 2735 \); \( 2735 + 900 = 3635 \); \( 3635 + 400 = 4035 \); \( 4035 + 820 = 4855 \); \( 4855 + 790 = 5645 \)
- \( \sum xy \):
\( 5\times25 = 125 \); \( 10\times80 = 800 \); \( 7\times90 = 630 \); \( 20\times220 = 4400 \); \( 30\times440 = 13200 \); \( 40\times520 = 20800 \); \( 50\times630 = 31500 \); \( 60\times730 = 43800 \); \( 70\times900 = 63000 \); \( 35\times400 = 14000 \); \( 55\times820 = 45100 \); \( 45\times790 = 35550 \)
Sum these: \( 125 + 800 = 925 \); \( 925 + 630 = 1555 \); \( 1555 + 4400 = 5955 \); \( 5955 + 13200 = 19155 \); \( 19155 + 20800 = 39955 \); \( 39955 + 31500 = 71455 \); \( 71455 + 43800 = 115255 \); \( 115255 + 63000 = 178255 \); \( 178255 + 14000 = 192255 \); \( 192255 + 45100 = 237355 \); \( 237355 + 35550 = 272905 \)
- \( \sum x^2 \):
\( 5^2 = 25 \); \( 10^2 = 100 \); \( 7^2 = 49 \); \( 20^2 = 400 \); \( 30^2 = 900 \); \( 40^2 = 1600 \); \( 50^2 = 2500 \); \( 60^2 = 3600 \); \( 70^2 = 4900 \); \( 35^2 = 1225 \); \( 55^2 = 3025 \); \( 45^2 = 2025 \)
Sum these: \( 25 + 100 = 125 \); \( 125 + 49 = 174 \); \( 174 + 400 = 574 \); \( 574 + 900 = 1474 \); \( 1474 + 1600 = 3074 \); \( 3074 + 2500 = 5574 \); \( 5574 + 3600 = 9174 \); \( 9174 + 4900 = 14074 \); \( 14074 + 1225 = 15299 \); \( 15299 + 3025 = 18324 \); \( 18324 + 2025 = 20349 \)
- \( \sum y^2 \):
\( 25^2 = 625 \); \( 80^2 = 6400 \); \( 90^2 = 8100 \); \( 220^2 = 48400 \); \( 440^2 = 193600 \); \( 520^2 = 270400 \); \( 630^2 = 396900 \); \( 730^2 = 532900 \); \( 900^2 = 810000 \); \( 400^2 = 160000 \); \( 820^2 = 672400 \); \( 790^2 = 624100 \)
Sum these: \( 625 + 6400 = 7025 \); \( 7025 + 8100 = 15125 \); \( 15125 + 48400 = 63525 \); \( 63525 + 193600 = 257125 \); \( 257125 + 270400 = 527525 \); \( 527525 + 396900 = 924425 \); \( 924425 + 532900 = 1457325 \); \( 1457325 + 810000 = 2267325 \); \( 2267325 + 160000 = 2427325 \); \( 2427325 + 672400 = 3099725 \); \( 3099725 + 624100 = 3723825 \)

Step3: Substitute into the formula

First, calculate the numerator:
\( n(\sum xy) - (\sum x)(\sum y) = 12\times272905 - 427\times5645 \)
\( 12\times272905 = 3274860 \)
\( 427\times5645 = 427\times(5000 + 600 + 45) = 427\times5000 + 427\times600 + 427\times45 = 2135000 + 256200 + 19215 = 2135000 + 256200 = 2391200 + 19215 = 2410415 \)
Numerator \( = 3274860 - 2410415 = 864445 \)

Calculate the denominator:
First, \( n\sum x^2 - (\sum x)^2 = 12\times20349 - 427^2 \)
\( 12\times20349 = 244188 \)
\( 427^2 = (400 + 27)^2 = 400^2 + 2\times400\times27 + 27^2 = 160000 + 21600 + 729 = 182329 \)
So \( 244188 - 182329 = 61859 \)

Next, \( n\sum y^2 - (\sum y)^2 = 12\times3723825 - 5645^2 \)
\( 12\times3723825 = 44685900 \)
\( 5645^2 = (5600 + 45)^2 = 5600^2 + 2\times5600\times45 + 45^2 = 31360000 + 504000 + 2025 = 31360000 + 504000 = 31864000 + 2025 = 31866025 \)
So \( 44685900 - 31866025 = 12819875 \)

Now, the denominator is \( \sqrt{61859\times12819875} \)
First, calculate \( 61859\times12819875 \approx 61859\times1.2819875\times10^7 \approx 793000000000 \) (approximate, but better to calculate directly)
\( 61859\times12819875 = 61859\times12819875 \)
Take square root: \( \sqrt{61859\times12819875} \approx \sqrt{793000000000} \approx 890500 \) (approximate, but let's use more accurate calculation)

Wait, actually, let's compute \( 61859\times12819875 \):

\( 61859\times12819875 = 61859\times(12000000 + 800000 + 19875) = 61859\times12000000 + 61859\times800000 + 61859\times19875 \)
\( = 742308000000 + 49487200000 + 1230000000 \) (approx)
\( = 742308000000 + 49487200000 = 791795200000 + 1230000000 = 793025200000 \)
Then \( \sqrt{793025200000} \approx 890519 \)

Now, \( r = \frac{864445}{890519} \approx 0.9707 \)

Answer:

0.9707