Question

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

x	y
30.9	70.6
29.4	54.8
29.9	63.4
34.1	78
54.3	67.1
-4.5	93.3

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:

• moderate positive correlation
• no relationship
• weak positive correlation
• strong negative correlation
• strong positive correlation
• weak negative correlation
• moderate negative correlation

scatter plot and regression equation calculator
https://www.socscistatistics.com/tests/regression/default.aspx
scatter plot and correlation coefficient calculator
https://www.socscistatistics.com/tests/pearson/default2.aspx
question help: message instructor

Explanation:

Step1: Recall the formula for Pearson's 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, \( \sum xy \) is the sum of the product of \( x \) and \( y \) values, \( \sum x \) is the sum of \( x \) values, \( \sum y \) is the sum of \( y \) values, \( \sum x^2 \) is the sum of the squares of \( x \) values, and \( \sum y^2 \) is the sum of the squares of \( y \) values.

First, we list out the data points:
\( (x_1, y_1) = (12.3, 89.1) \)
\( (x_2, y_2) = (30.9, 70.6) \)
\( (x_3, y_3) = (29.4, 54.8) \)
\( (x_4, y_4) = (29.9, 63.4) \)
\( (x_5, y_5) = (34.1, 78.0) \)
\( (x_6, y_6) = (54.3, 67.1) \)
\( (x_7, y_7) = (-4.5, 93.3) \)
So, \( n = 7 \).

Step2: Calculate \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), \( \sum y^2 \)

- Calculate \( \sum x \):
\[ \sum x = 12.3 + 30.9 + 29.4 + 29.9 + 34.1 + 54.3 - 4.5 \]
\[ \sum x = 12.3 + 30.9 = 43.2; 43.2 + 29.4 = 72.6; 72.6 + 29.9 = 102.5; 102.5 + 34.1 = 136.6; 136.6 + 54.3 = 190.9; 190.9 - 4.5 = 186.4 \]
- Calculate \( \sum y \):
\[ \sum y = 89.1 + 70.6 + 54.8 + 63.4 + 78.0 + 67.1 + 93.3 \]
\[ 89.1 + 70.6 = 159.7; 159.7 + 54.8 = 214.5; 214.5 + 63.4 = 277.9; 277.9 + 78.0 = 355.9; 355.9 + 67.1 = 423.0; 423.0 + 93.3 = 516.3 \]
- Calculate \( \sum xy \):
\[ \begin{align*} (12.3)(89.1) + (30.9)(70.6) + (29.4)(54.8) + (29.9)(63.4) + (34.1)(78.0) + (54.3)(67.1) + (-4.5)(93.3)\\ = 1095.93 + 2181.54 + 1611.12 + 1895.66 + 2659.8 + 3643.53 - 419.85\\ \end{align*} \]
Calculating each term:
\( 12.3\times89.1 = 1095.93 \)
\( 30.9\times70.6 = 2181.54 \)
\( 29.4\times54.8 = 1611.12 \)
\( 29.9\times63.4 = 1895.66 \)
\( 34.1\times78.0 = 2659.8 \)
\( 54.3\times67.1 = 3643.53 \)
\( -4.5\times93.3 = -419.85 \)
Now sum these:
\( 1095.93 + 2181.54 = 3277.47 \)
\( 3277.47 + 1611.12 = 4888.59 \)
\( 4888.59 + 1895.66 = 6784.25 \)
\( 6784.25 + 2659.8 = 9444.05 \)
\( 9444.05 + 3643.53 = 13087.58 \)
\( 13087.58 - 419.85 = 12667.73 \)
So, \( \sum xy = 12667.73 \)
- Calculate \( \sum x^2 \):
\[ \begin{align*} (12.3)^2 + (30.9)^2 + (29.4)^2 + (29.9)^2 + (34.1)^2 + (54.3)^2 + (-4.5)^2\\ = 151.29 + 954.81 + 864.36 + 894.01 + 1162.81 + 2948.49 + 20.25\\ \end{align*} \]
Calculating each term:
\( 12.3^2 = 151.29 \)
\( 30.9^2 = 954.81 \)
\( 29.4^2 = 864.36 \)
\( 29.9^2 = 894.01 \)
\( 34.1^2 = 1162.81 \)
\( 54.3^2 = 2948.49 \)
\( (-4.5)^2 = 20.25 \)
Summing these:
\( 151.29 + 954.81 = 1106.1 \)
\( 1106.1 + 864.36 = 1970.46 \)
\( 1970.46 + 894.01 = 2864.47 \)
\( 2864.47 + 1162.81 = 4027.28 \)
\( 4027.28 + 2948.49 = 6975.77 \)
\( 6975.77 + 20.25 = 6996.02 \)
So, \( \sum x^2 = 6996.02 \)
- Calculate \( \sum y^2 \):
\[ \begin{align*} (89.1)^2 + (70.6)^2 + (54.8)^2 + (63.4)^2 + (78.0)^2 + (67.1)^2 + (93.3)^2\\ = 7938.81 + 4984.36 + 3003.04 + 4019.56 + 6084.0 + 4502.41 + 8704.89\\ \end{align*} \]
Calculating each term:
\( 89.1^2 = 7938.81 \)
\( 70.6^2 = 4984.36 \)
\( 54.8^2 = 3003.04 \)
\( 63.4^2 = 4019.56 \)
\( 78.0^2 = 6084.0 \)
\( 67.1^2 = 4502.41 \)
\( 93.3^2 = 8704.89 \)
Summing these:
\( 7938.81 + 4984.36 = 12923.17 \)
\( 12923.17 + 3003.04 = 15926.21 \)
\( 15926.21 + 4019.56 = 19945.77 \)
\( 19945.77 + 6084.0 = 26029.77 \)
\( 26029.77 + 4502.41 = 30532.18 \)
\( 30532.18 + 8704.89 = 39237.07 \)
So, \( \sum y^2 = 39237.07 \)

Step3: Substitute the values into the formula

We have \( n = 7 \), \( \sum x = 186.4 \), \( \sum y = 516.3 \), \( \sum xy = 12667.73 \), \( \sum x^2 = 6996.02 \), \( \sum y^2 = 39237.07 \)

First, calculate the numerator:
\[ n(\sum xy) - (\sum x)(\sum y) = 7(12667.73) - (186.4)(516.3) \]
\[ 7\times12667.73 = 88674.11 \]
\[ 186.4\times516.3 = 186.4\times500 + 186.4\times16.3 = 93200 + 3038.32 = 96238.32 \]
\[ \text{Numerator} = 88674.11 - 96238.32 = -7564.21 \]

Next, calculate the denominator:
First, calculate \( n\sum x^2 - (\sum x)^2 \):
\[ 7\times6996.02 - (186.4)^2 = 48972.14 - 34744.96 = 14227.18 \]
Then, calculate \( n\sum y^2 - (\sum y)^2 \):
\[ 7\times39237.07 - (516.3)^2 = 274659.49 - 266565.69 = 8093.8 \]
Now, multiply these two results:
\[ 14227.18\times8093.8 \approx 14227.18\times8000 + 14227.18\times93.8 \approx 113817440 + 1334500.484 = 115151940.484 \]
Take the square root:
\[ \sqrt{115151940.484} \approx 10731.0 \] (We can also calculate more accurately, but let's proceed)

Now, the denominator is \( \sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]} = \sqrt{14227.18\times8093.8} \)
\[ \sqrt{14227.18\times8093.8} \approx \sqrt{115151940.484} \approx 10731.0 \] (A more accurate calculation: \( 14227.18\times8093.8 = 14227.18\times8000 + 14227.18\times93.8 = 113817440 + 1334500.484 = 115151940.484 \), square root of that is approximately \( \sqrt{115151940.484} \approx 10731.0 \))

Now, the correlation coefficient \( r \) is:
\[ r = \frac{-7564.21}{10731.0} \approx -0.705 \] (Wait, let's check the calculations again for accuracy. Maybe we made a mistake in multiplication. Let's recalculate the denominator more accurately.)

Wait, actually, \( 14227.18\times8093.8 \):
Let's compute \( 14227.18\times8093.8 \) as:
\( 14227.18\times8093.8 = 14227.18\times(8000 + 93.8) = 14227.18\times8000 + 14227.18\times93.8 \)
\( 14227.18\times8000 = 113817440 \)
\( 14227.18\times93.8 = 14227.18\times(90 + 3.8) = 14227.18\times90 + 14227.18\times3.8 = 1280446.2 + 54063.284 = 1334509.484 \)
So total is \( 113817440 + 1334509.484 = 115151949.484 \)
Square root of \( 115151949.484 \) is \( \sqrt{115151949.484} \approx 10731.0 \) (since \( 10731^2 = (10000 + 731)^2 = 100000000 + 2\times10000\times731 + 731^2 = 100000000 + 14620000 + 534361 = 115154361 \), which is close to 115151949.484, so maybe a miscalculation earlier. Let's use a better approach.

Wait, maybe we should use a calculator for more precision. Alternatively, let's use the formula step by step with more precise calculations.

Alternatively, use the formula for \( r \) using a calculator-like approach. Let's recalculate the numerator and denominator with more precision.

Numerator: \( 7*12667.73 - 186.4*516.3 \)
\( 7*12667.73 = 88674.11 \)
\( 186.4*516.3 \): Let's calculate 186.4*516.3
186.4 * 500 = 93200
186.4 * 16 = 2982.4
186.4 * 0.3 = 55.92
So 186.4*516.3 = 93200 + 2982.4 + 55.92 = 96238.32
So numerator = 88674.11 - 96238.32 = -7564.21

Denominator part 1: \( 7*6996.02 - 186.4^2 \)
7*6996.02 = 4

Answer:

Step1: Recall the formula for Pearson's 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, \( \sum xy \) is the sum of the product of \( x \) and \( y \) values, \( \sum x \) is the sum of \( x \) values, \( \sum y \) is the sum of \( y \) values, \( \sum x^2 \) is the sum of the squares of \( x \) values, and \( \sum y^2 \) is the sum of the squares of \( y \) values.

First, we list out the data points:
\( (x_1, y_1) = (12.3, 89.1) \)
\( (x_2, y_2) = (30.9, 70.6) \)
\( (x_3, y_3) = (29.4, 54.8) \)
\( (x_4, y_4) = (29.9, 63.4) \)
\( (x_5, y_5) = (34.1, 78.0) \)
\( (x_6, y_6) = (54.3, 67.1) \)
\( (x_7, y_7) = (-4.5, 93.3) \)
So, \( n = 7 \).

Step2: Calculate \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), \( \sum y^2 \)

• Calculate \( \sum x \):

$$ \sum x = 12.3 + 30.9 + 29.4 + 29.9 + 34.1 + 54.3 - 4.5 $$

$$ \sum x = 12.3 + 30.9 = 43.2; 43.2 + 29.4 = 72.6; 72.6 + 29.9 = 102.5; 102.5 + 34.1 = 136.6; 136.6 + 54.3 = 190.9; 190.9 - 4.5 = 186.4 $$

• Calculate \( \sum y \):

$$ \sum y = 89.1 + 70.6 + 54.8 + 63.4 + 78.0 + 67.1 + 93.3 $$

$$ 89.1 + 70.6 = 159.7; 159.7 + 54.8 = 214.5; 214.5 + 63.4 = 277.9; 277.9 + 78.0 = 355.9; 355.9 + 67.1 = 423.0; 423.0 + 93.3 = 516.3 $$

• Calculate \( \sum xy \):

$$
LATEXBLOCK0
$$

Calculating each term:
\( 12.3\times89.1 = 1095.93 \)
\( 30.9\times70.6 = 2181.54 \)
\( 29.4\times54.8 = 1611.12 \)
\( 29.9\times63.4 = 1895.66 \)
\( 34.1\times78.0 = 2659.8 \)
\( 54.3\times67.1 = 3643.53 \)
\( -4.5\times93.3 = -419.85 \)
Now sum these:
\( 1095.93 + 2181.54 = 3277.47 \)
\( 3277.47 + 1611.12 = 4888.59 \)
\( 4888.59 + 1895.66 = 6784.25 \)
\( 6784.25 + 2659.8 = 9444.05 \)
\( 9444.05 + 3643.53 = 13087.58 \)
\( 13087.58 - 419.85 = 12667.73 \)
So, \( \sum xy = 12667.73 \)

• Calculate \( \sum x^2 \):

$$
LATEXBLOCK1
$$

Calculating each term:
\( 12.3^2 = 151.29 \)
\( 30.9^2 = 954.81 \)
\( 29.4^2 = 864.36 \)
\( 29.9^2 = 894.01 \)
\( 34.1^2 = 1162.81 \)
\( 54.3^2 = 2948.49 \)
\( (-4.5)^2 = 20.25 \)
Summing these:
\( 151.29 + 954.81 = 1106.1 \)
\( 1106.1 + 864.36 = 1970.46 \)
\( 1970.46 + 894.01 = 2864.47 \)
\( 2864.47 + 1162.81 = 4027.28 \)
\( 4027.28 + 2948.49 = 6975.77 \)
\( 6975.77 + 20.25 = 6996.02 \)
So, \( \sum x^2 = 6996.02 \)

• Calculate \( \sum y^2 \):

$$
LATEXBLOCK2
$$

Calculating each term:
\( 89.1^2 = 7938.81 \)
\( 70.6^2 = 4984.36 \)
\( 54.8^2 = 3003.04 \)
\( 63.4^2 = 4019.56 \)
\( 78.0^2 = 6084.0 \)
\( 67.1^2 = 4502.41 \)
\( 93.3^2 = 8704.89 \)
Summing these:
\( 7938.81 + 4984.36 = 12923.17 \)
\( 12923.17 + 3003.04 = 15926.21 \)
\( 15926.21 + 4019.56 = 19945.77 \)
\( 19945.77 + 6084.0 = 26029.77 \)
\( 26029.77 + 4502.41 = 30532.18 \)
\( 30532.18 + 8704.89 = 39237.07 \)
So, \( \sum y^2 = 39237.07 \)

Step3: Substitute the values into the formula

We have \( n = 7 \), \( \sum x = 186.4 \), \( \sum y = 516.3 \), \( \sum xy = 12667.73 \), \( \sum x^2 = 6996.02 \), \( \sum y^2 = 39237.07 \)

First, calculate the numerator:

$$ n(\sum xy) - (\sum x)(\sum y) = 7(12667.73) - (186.4)(516.3) $$

$$ 7\times12667.73 = 88674.11 $$

$$ 186.4\times516.3 = 186.4\times500 + 186.4\times16.3 = 93200 + 3038.32 = 96238.32 $$

$$ \text{Numerator} = 88674.11 - 96238.32 = -7564.21 $$

Next, calculate the denominator:
First, calculate \( n\sum x^2 - (\sum x)^2 \):

$$ 7\times6996.02 - (186.4)^2 = 48972.14 - 34744.96 = 14227.18 $$

Then, calculate \( n\sum y^2 - (\sum y)^2 \):

$$ 7\times39237.07 - (516.3)^2 = 274659.49 - 266565.69 = 8093.8 $$

Now, multiply these two results:

$$ 14227.18\times8093.8 \approx 14227.18\times8000 + 14227.18\times93.8 \approx 113817440 + 1334500.484 = 115151940.484 $$

Take the square root:

$$ \sqrt{115151940.484} \approx 10731.0 $$

(We can also calculate more accurately, but let's proceed)

Now, the denominator is \( \sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]} = \sqrt{14227.18\times8093.8} \)

$$ \sqrt{14227.18\times8093.8} \approx \sqrt{115151940.484} \approx 10731.0 $$

(A more accurate calculation: \( 14227.18\times8093.8 = 14227.18\times8000 + 14227.18\times93.8 = 113817440 + 1334500.484 = 115151940.484 \), square root of that is approximately \( \sqrt{115151940.484} \approx 10731.0 \))

Now, the correlation coefficient \( r \) is:

$$ r = \frac{-7564.21}{10731.0} \approx -0.705 $$

(Wait, let's check the calculations again for accuracy. Maybe we made a mistake in multiplication. Let's recalculate the denominator more accurately.)

Wait, actually, \( 14227.18\times8093.8 \):
Let's compute \( 14227.18\times8093.8 \) as:
\( 14227.18\times8093.8 = 14227.18\times(8000 + 93.8) = 14227.18\times8000 + 14227.18\times93.8 \)
\( 14227.18\times8000 = 113817440 \)
\( 14227.18\times93.8 = 14227.18\times(90 + 3.8) = 14227.18\times90 + 14227.18\times3.8 = 1280446.2 + 54063.284 = 1334509.484 \)
So total is \( 113817440 + 1334509.484 = 115151949.484 \)
Square root of \( 115151949.484 \) is \( \sqrt{115151949.484} \approx 10731.0 \) (since \( 10731^2 = (10000 + 731)^2 = 100000000 + 2\times10000\times731 + 731^2 = 100000000 + 14620000 + 534361 = 115154361 \), which is close to 115151949.484, so maybe a miscalculation earlier. Let's use a better approach.

Wait, maybe we should use a calculator for more precision. Alternatively, let's use the formula step by step with more precise calculations.

Alternatively, use the formula for \( r \) using a calculator-like approach. Let's recalculate the numerator and denominator with more precision.

Numerator: \( 7*12667.73 - 186.4*516.3 \)
\( 7*12667.73 = 88674.11 \)
\( 186.4*516.3 \): Let's calculate 186.4*516.3
186.4 * 500 = 93200
186.4 * 16 = 2982.4
186.4 * 0.3 = 55.92
So 186.4*516.3 = 93200 + 2982.4 + 55.92 = 96238.32
So numerator = 88674.11 - 96238.32 = -7564.21

Denominator part 1: \( 7*6996.02 - 186.4^2 \)
7*6996.02 = 4