Question

given the following bivariate data set, find the correlation coefficient and describe the linear relationship.

x	y
21.6	58
22.2	60.6
35.5	83.7
28.3	58
26.3	72.7
33	54.5

a) find the correlation coefficient and round to three decimal places.
$r \approx \square$
b) what type of linear relationship is there between the variables?
the linear relationship is:
\\(\circ\\) no relationship \\(\circ\\) strong positive correlation
\\(\circ\\) weak positive correlation \\(\circ\\) weak negative correlation
\\(\circ\\) strong negative correlation \\(\circ\\) moderate negative correlation
\\(\circ\\) moderate positive correlation

Explanation:

Step 1: List the data points

We have the following pairs of (x, y) values: (14.1, 79.2), (21.6, 58), (22.2, 60.6), (35.5, 83.7), (28.3, 58), (26.3, 72.7), (33, 54.5). First, we need to calculate the necessary sums for the correlation coefficient formula. The formula for the 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. Here, \( n = 7 \).

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

- Calculate \( \sum x \):
\( 14.1 + 21.6 + 22.2 + 35.5 + 28.3 + 26.3 + 33 = 181 \) (let's verify: 14.1+21.6=35.7; 35.7+22.2=57.9; 57.9+35.5=93.4; 93.4+28.3=121.7; 121.7+26.3=148; 148+33=181. Correct.)
- Calculate \( \sum y \):
\( 79.2 + 58 + 60.6 + 83.7 + 58 + 72.7 + 54.5 = 466.7 \) (Verify: 79.2+58=137.2; 137.2+60.6=197.8; 197.8+83.7=281.5; 281.5+58=339.5; 339.5+72.7=412.2; 412.2+54.5=466.7. Correct.)
- Calculate \( \sum xy \):
For each pair:
\( 14.1\times79.2 = 1116.72 \)
\( 21.6\times58 = 1252.8 \)
\( 22.2\times60.6 = 1345.32 \)
\( 35.5\times83.7 = 2971.35 \)
\( 28.3\times58 = 1641.4 \)
\( 26.3\times72.7 = 1912.01 \)
\( 33\times54.5 = 1808.5 \)
Now sum these: \( 1116.72 + 1252.8 + 1345.32 + 2971.35 + 1641.4 + 1912.01 + 1808.5 \)
Let's add step by step:
1116.72 + 1252.8 = 2369.52
2369.52 + 1345.32 = 3714.84
3714.84 + 2971.35 = 6686.19
6686.19 + 1641.4 = 8327.59
8327.59 + 1912.01 = 10239.6
10239.6 + 1808.5 = 12048.1
So \( \sum xy = 12048.1 \)
- Calculate \( \sum x^2 \):
\( 14.1^2 = 198.81 \)
\( 21.6^2 = 466.56 \)
\( 22.2^2 = 492.84 \)
\( 35.5^2 = 1260.25 \)
\( 28.3^2 = 800.89 \)
\( 26.3^2 = 691.69 \)
\( 33^2 = 1089 \)
Sum these: \( 198.81 + 466.56 + 492.84 + 1260.25 + 800.89 + 691.69 + 1089 \)
Step by step:
198.81 + 466.56 = 665.37
665.37 + 492.84 = 1158.21
1158.21 + 1260.25 = 2418.46
2418.46 + 800.89 = 3219.35
3219.35 + 691.69 = 3911.04
3911.04 + 1089 = 5000.04
So \( \sum x^2 = 5000.04 \)
- Calculate \( \sum y^2 \):
\( 79.2^2 = 6272.64 \)
\( 58^2 = 3364 \)
\( 60.6^2 = 3672.36 \)
\( 83.7^2 = 7005.69 \)
\( 58^2 = 3364 \)
\( 72.7^2 = 5285.29 \)
\( 54.5^2 = 2970.25 \)
Sum these: \( 6272.64 + 3364 + 3672.36 + 7005.69 + 3364 + 5285.29 + 2970.25 \)
Step by step:
6272.64 + 3364 = 9636.64
9636.64 + 3672.36 = 13309
13309 + 7005.69 = 20314.69
20314.69 + 3364 = 23678.69
23678.69 + 5285.29 = 28963.98
28963.98 + 2970.25 = 31934.23
So \( \sum y^2 = 31934.23 \)

Step 3: Plug into the formula

First, calculate the numerator: \( n\sum xy - \sum x \sum y = 7\times12048.1 - 181\times466.7 \)
Calculate \( 7\times12048.1 = 84336.7 \)
Calculate \( 181\times466.7 = 181\times466.7 \). Let's compute 180×466.7=84006, 1×466.7=466.7, so total is 84006 + 466.7 = 84472.7
Numerator: \( 84336.7 - 84472.7 = -136 \)

Now calculate the denominator: \( \sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]} \)
First, \( n\sum x^2 - (\sum x)^2 = 7\times5000.04 - 181^2 \)
\( 7\times5000.04 = 35000.28 \)
\( 181^2 = 32761 \)
So \( 35000.28 - 32761 = 2239.28 \)

Next, \( n\sum y^2 - (\sum y)^2 = 7\times31934.23 - 466.7^2 \)
\( 7\times31934.23 = 223539.61 \)
\( 466.7^2 \): Let's calculate 460^2=211600, 6.7^2=44.89, 2×460×6.7=6164, so (460+6.7)^2=460^2 + 2×460×6.7 + 6.7^2=211600 + 6164 + 44.89=217808.89
So \( 223539.61 - 217808.89 = 5730.72 \)

Now multiply these two results: \( 2239.28 \times 5730.72 \approx 2239.28\times5730.72 \). Let's compute the square root of this product. Wait, actually, the denominator is the square root of (2239.28 × 5730.72). Let's first compute 2239.28 × 5730.72:

2239.28 × 5730.72 ≈ 2239.28 × 5730 ≈ 2239.28×5000=11,196,400; 2239.28×730=2239.28×700=1,567,496; 2239.28×30=67,178.4; so 1,567,496 + 67,178.4=1,634,674.4; total 11,196,400 + 1,634,674.4=12,831,074.4. Then multiply by 0.72: 12,831,074.4×0.72≈9,238,373.568. So total ≈12,831,074.4 + 9,238,373.568≈22,069,447.968. Now take the square root: \( \sqrt{22069447.968} \approx 4697.813 \) (Wait, maybe we made a mistake in calculation. Let's check the sums again, because the numerator is -136, which is small, but let's recalculate the sums more accurately.)

Wait, let's recalculate \( \sum x \):

14.1 + 21.6 = 35.7

35.7 + 22.2 = 57.9

57.9 + 35.5 = 93.4

93.4 + 28.3 = 121.7

121.7 + 26.3 = 148

148 + 33 = 181. Correct.

\( \sum y \):

79.2 + 58 = 137.2

137.2 + 60.6 = 197.8

197.8 + 83.7 = 281.5

281.5 + 58 = 339.5

339.5 + 72.7 = 412.2

412.2 + 54.5 = 466.7. Correct.

\( \sum xy \):

14.1×79.2: 14×79.2=1108.8, 0.1×79.2=7.92, total 1108.8+7.92=1116.72. Correct.

21.6×58: 20×58=1160, 1.6×58=92.8, total 1160+92.8=1252.8. Correct.

22.2×60.6: 22×60.6=1333.2, 0.2×60.6=12.12, total 1333.2+12.12=1345.32. Correct.

35.5×83.7: 35×83.7=2929.5, 0.5×83.7=41.85, total 2929.5+41.85=2971.35. Correct.

28.3×58: 28×58=1624, 0.3×58=17.4, total 1624+17.4=1641.4. Correct.

26.3×72.7: 26×72.7=1890.2, 0.3×72.7=21.81, total 1890.2+21.81=1912.01. Correct.

33×54.5: 30×54.5=1635, 3×54.5=163.5, total 1635+163.5=1808.5. Correct.

Sum of xy: 1116.72 + 1252.8 = 2369.52; +1345.32=3714.84; +2971.35=6686.19; +1641.4=8327.59; +1912.01=10239.6; +1808.5=12048.1. Correct.

\( \sum x^2 \):

14.1²=198.81, 21.6²=466.56, 22.2²=492.84, 35.5²=1260.25, 28.3²=800.89, 26.3²=691.69, 33²=1089. Sum: 198.81+466.56=665.37; +492.84=1158.21; +1260.25=2418.46; +800.89=3219.35; +691.69=3911.04; +1089=5000.04. Correct.

\( \sum y^2 \):

79.2²=6272.64, 58²=3364, 60.6²=3672.36, 83.7²=7005.69, 58²=3364, 72.7²=5285.29, 54.5²=2970.25. Sum: 6272.64+3364=9636.64; +3672.36=13309; +7005.69=20314.69; +3364=23678.69; +5285.29=28963.98; +2970.25=31934.23. Correct.

Now numerator: 7*12048.1 - 181*466.7 = 84336.7 - (181*466.7). Let's calculate 181*466.7:

466.7 * 180 = 84006, 466.7

Answer:

Step 1: List the data points

We have the following pairs of (x, y) values: (14.1, 79.2), (21.6, 58), (22.2, 60.6), (35.5, 83.7), (28.3, 58), (26.3, 72.7), (33, 54.5). First, we need to calculate the necessary sums for the correlation coefficient formula. The formula for the 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. Here, \( n = 7 \).

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

• Calculate \( \sum x \):

\( 14.1 + 21.6 + 22.2 + 35.5 + 28.3 + 26.3 + 33 = 181 \) (let's verify: 14.1+21.6=35.7; 35.7+22.2=57.9; 57.9+35.5=93.4; 93.4+28.3=121.7; 121.7+26.3=148; 148+33=181. Correct.)

• Calculate \( \sum y \):

\( 79.2 + 58 + 60.6 + 83.7 + 58 + 72.7 + 54.5 = 466.7 \) (Verify: 79.2+58=137.2; 137.2+60.6=197.8; 197.8+83.7=281.5; 281.5+58=339.5; 339.5+72.7=412.2; 412.2+54.5=466.7. Correct.)

• Calculate \( \sum xy \):

For each pair:
\( 14.1\times79.2 = 1116.72 \)
\( 21.6\times58 = 1252.8 \)
\( 22.2\times60.6 = 1345.32 \)
\( 35.5\times83.7 = 2971.35 \)
\( 28.3\times58 = 1641.4 \)
\( 26.3\times72.7 = 1912.01 \)
\( 33\times54.5 = 1808.5 \)
Now sum these: \( 1116.72 + 1252.8 + 1345.32 + 2971.35 + 1641.4 + 1912.01 + 1808.5 \)
Let's add step by step:
1116.72 + 1252.8 = 2369.52
2369.52 + 1345.32 = 3714.84
3714.84 + 2971.35 = 6686.19
6686.19 + 1641.4 = 8327.59
8327.59 + 1912.01 = 10239.6
10239.6 + 1808.5 = 12048.1
So \( \sum xy = 12048.1 \)

• Calculate \( \sum x^2 \):

\( 14.1^2 = 198.81 \)
\( 21.6^2 = 466.56 \)
\( 22.2^2 = 492.84 \)
\( 35.5^2 = 1260.25 \)
\( 28.3^2 = 800.89 \)
\( 26.3^2 = 691.69 \)
\( 33^2 = 1089 \)
Sum these: \( 198.81 + 466.56 + 492.84 + 1260.25 + 800.89 + 691.69 + 1089 \)
Step by step:
198.81 + 466.56 = 665.37
665.37 + 492.84 = 1158.21
1158.21 + 1260.25 = 2418.46
2418.46 + 800.89 = 3219.35
3219.35 + 691.69 = 3911.04
3911.04 + 1089 = 5000.04
So \( \sum x^2 = 5000.04 \)

• Calculate \( \sum y^2 \):

\( 79.2^2 = 6272.64 \)
\( 58^2 = 3364 \)
\( 60.6^2 = 3672.36 \)
\( 83.7^2 = 7005.69 \)
\( 58^2 = 3364 \)
\( 72.7^2 = 5285.29 \)
\( 54.5^2 = 2970.25 \)
Sum these: \( 6272.64 + 3364 + 3672.36 + 7005.69 + 3364 + 5285.29 + 2970.25 \)
Step by step:
6272.64 + 3364 = 9636.64
9636.64 + 3672.36 = 13309
13309 + 7005.69 = 20314.69
20314.69 + 3364 = 23678.69
23678.69 + 5285.29 = 28963.98
28963.98 + 2970.25 = 31934.23
So \( \sum y^2 = 31934.23 \)

Step 3: Plug into the formula

First, calculate the numerator: \( n\sum xy - \sum x \sum y = 7\times12048.1 - 181\times466.7 \)
Calculate \( 7\times12048.1 = 84336.7 \)
Calculate \( 181\times466.7 = 181\times466.7 \). Let's compute 180×466.7=84006, 1×466.7=466.7, so total is 84006 + 466.7 = 84472.7
Numerator: \( 84336.7 - 84472.7 = -136 \)

Now calculate the denominator: \( \sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]} \)
First, \( n\sum x^2 - (\sum x)^2 = 7\times5000.04 - 181^2 \)
\( 7\times5000.04 = 35000.28 \)
\( 181^2 = 32761 \)
So \( 35000.28 - 32761 = 2239.28 \)

Next, \( n\sum y^2 - (\sum y)^2 = 7\times31934.23 - 466.7^2 \)
\( 7\times31934.23 = 223539.61 \)
\( 466.7^2 \): Let's calculate 460^2=211600, 6.7^2=44.89, 2×460×6.7=6164, so (460+6.7)^2=460^2 + 2×460×6.7 + 6.7^2=211600 + 6164 + 44.89=217808.89
So \( 223539.61 - 217808.89 = 5730.72 \)

Now multiply these two results: \( 2239.28 \times 5730.72 \approx 2239.28\times5730.72 \). Let's compute the square root of this product. Wait, actually, the denominator is the square root of (2239.28 × 5730.72). Let's first compute 2239.28 × 5730.72:

2239.28 × 5730.72 ≈ 2239.28 × 5730 ≈ 2239.28×5000=11,196,400; 2239.28×730=2239.28×700=1,567,496; 2239.28×30=67,178.4; so 1,567,496 + 67,178.4=1,634,674.4; total 11,196,400 + 1,634,674.4=12,831,074.4. Then multiply by 0.72: 12,831,074.4×0.72≈9,238,373.568. So total ≈12,831,074.4 + 9,238,373.568≈22,069,447.968. Now take the square root: \( \sqrt{22069447.968} \approx 4697.813 \) (Wait, maybe we made a mistake in calculation. Let's check the sums again, because the numerator is -136, which is small, but let's recalculate the sums more accurately.)

Wait, let's recalculate \( \sum x \):

14.1 + 21.6 = 35.7

35.7 + 22.2 = 57.9

57.9 + 35.5 = 93.4

93.4 + 28.3 = 121.7

121.7 + 26.3 = 148

148 + 33 = 181. Correct.

\( \sum y \):

79.2 + 58 = 137.2

137.2 + 60.6 = 197.8

197.8 + 83.7 = 281.5

281.5 + 58 = 339.5

339.5 + 72.7 = 412.2

412.2 + 54.5 = 466.7. Correct.

\( \sum xy \):

14.1×79.2: 14×79.2=1108.8, 0.1×79.2=7.92, total 1108.8+7.92=1116.72. Correct.

21.6×58: 20×58=1160, 1.6×58=92.8, total 1160+92.8=1252.8. Correct.

22.2×60.6: 22×60.6=1333.2, 0.2×60.6=12.12, total 1333.2+12.12=1345.32. Correct.

35.5×83.7: 35×83.7=2929.5, 0.5×83.7=41.85, total 2929.5+41.85=2971.35. Correct.

28.3×58: 28×58=1624, 0.3×58=17.4, total 1624+17.4=1641.4. Correct.

26.3×72.7: 26×72.7=1890.2, 0.3×72.7=21.81, total 1890.2+21.81=1912.01. Correct.

33×54.5: 30×54.5=1635, 3×54.5=163.5, total 1635+163.5=1808.5. Correct.

Sum of xy: 1116.72 + 1252.8 = 2369.52; +1345.32=3714.84; +2971.35=6686.19; +1641.4=8327.59; +1912.01=10239.6; +1808.5=12048.1. Correct.

\( \sum x^2 \):

14.1²=198.81, 21.6²=466.56, 22.2²=492.84, 35.5²=1260.25, 28.3²=800.89, 26.3²=691.69, 33²=1089. Sum: 198.81+466.56=665.37; +492.84=1158.21; +1260.25=2418.46; +800.89=3219.35; +691.69=3911.04; +1089=5000.04. Correct.

\( \sum y^2 \):

79.2²=6272.64, 58²=3364, 60.6²=3672.36, 83.7²=7005.69, 58²=3364, 72.7²=5285.29, 54.5²=2970.25. Sum: 6272.64+3364=9636.64; +3672.36=13309; +7005.69=20314.69; +3364=23678.69; +5285.29=28963.98; +2970.25=31934.23. Correct.

Now numerator: 712048.1 - 181466.7 = 84336.7 - (181466.7). Let's calculate 181466.7:

466.7 * 180 = 84006, 466.7