Question

1. using the correlation coefficient, what is the relationship between the variables?

x: 90, 87, 83, 92, 85
y: 80, 89, 94, 82, 91
options: weak, strong

Explanation:

Step1: List the data points

We have the following pairs of \( (x, y) \) values: \( (90, 80) \), \( (87, 89) \), \( (83, 94) \), \( (92, 82) \), \( (85, 91) \).

Step2: Calculate the necessary sums

First, we calculate the mean of \( x \) and \( y \).

The sum of \( x \) values: \( \sum x = 90 + 87 + 83 + 92 + 85 = 437 \)
The mean of \( x \), \( \bar{x} = \frac{437}{5} = 87.4 \)

The sum of \( y \) values: \( \sum y = 80 + 89 + 94 + 82 + 91 = 436 \)
The mean of \( y \), \( \bar{y} = \frac{436}{5} = 87.2 \)

Now, we calculate the numerator (covariance) and the denominator (product of standard deviations) for the correlation coefficient \( r \).

For the numerator:
\( \sum (x_i - \bar{x})(y_i - \bar{y}) \)

Calculating each term:
- For \( (90, 80) \): \( (90 - 87.4)(80 - 87.2) = (2.6)(-7.2) = -18.72 \)
- For \( (87, 89) \): \( (87 - 87.4)(89 - 87.2) = (-0.4)(1.8) = -0.72 \)
- For \( (83, 94) \): \( (83 - 87.4)(94 - 87.2) = (-4.4)(6.8) = -29.92 \)
- For \( (92, 82) \): \( (92 - 87.4)(82 - 87.2) = (4.6)(-5.2) = -23.92 \)
- For \( (85, 91) \): \( (85 - 87.4)(91 - 87.2) = (-2.4)(3.8) = -9.12 \)

Summing these terms: \( -18.72 - 0.72 - 29.92 - 23.92 - 9.12 = -82.4 \)

For the denominator:
First, calculate the standard deviation of \( x \), \( s_x \):
\( \sum (x_i - \bar{x})^2 \)
- \( (90 - 87.4)^2 = 6.76 \)
- \( (87 - 87.4)^2 = 0.16 \)
- \( (83 - 87.4)^2 = 19.36 \)
- \( (92 - 87.4)^2 = 21.16 \)
- \( (85 - 87.4)^2 = 5.76 \)
Summing these: \( 6.76 + 0.16 + 19.36 + 21.16 + 5.76 = 53.2 \)
\( s_x = \sqrt{\frac{53.2}{4}} = \sqrt{13.3} \approx 3.647 \) (we divide by \( n - 1 = 4 \) for sample standard deviation)

Next, calculate the standard deviation of \( y \), \( s_y \):
\( \sum (y_i - \bar{y})^2 \)
- \( (80 - 87.2)^2 = 51.84 \)
- \( (89 - 87.2)^2 = 3.24 \)
- \( (94 - 87.2)^2 = 46.24 \)
- \( (82 - 87.2)^2 = 27.04 \)
- \( (91 - 87.2)^2 = 14.44 \)
Summing these: \( 51.84 + 3.24 + 46.24 + 27.04 + 14.44 = 142.8 \)
\( s_y = \sqrt{\frac{142.8}{4}} = \sqrt{35.7} \approx 5.975 \)

Now, the denominator is \( s_x \times s_y \approx 3.647 \times 5.975 \approx 21.80 \)

Step3: Calculate the correlation coefficient \( r \)

\( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n - 1)s_x s_y} \) (using sample formula)
We have \( \sum (x_i - \bar{x})(y_i - \bar{y}) = -82.4 \) and \( (n - 1)s_x s_y = 4 \times 21.80 \approx 87.2 \) (Wait, no, actually the formula for the sample correlation coefficient is \( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \))

Let's recalculate the denominator correctly: \( \sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2} = \sqrt{53.2 \times 142.8} = \sqrt{7596.96} \approx 87.16 \)

So, \( r = \frac{-82.4}{87.16} \approx -0.945 \)

The absolute value of \( r \) is close to 1, which indicates a strong relationship. However, the negative sign indicates a negative correlation, but the magnitude is strong.

But wait, maybe I made a mistake in calculation. Let's check the data again. Wait, the \( x \) and \( y \) values: when \( x \) decreases, \( y \) increases? Let's see:

\( x = 90 \), \( y = 80 \); \( x = 87 \), \( y = 89 \); \( x = 83 \), \( y = 94 \); \( x = 92 \), \( y = 82 \); \( x = 85 \), \( y = 91 \)

So as \( x \) decreases from 90 to 87 to 83, \( y \) increases from 80 to 89 to 94. Then \( x \) increases to 92, \( y \) decreases to 82, and \( x \) increases to 85 (wait, 85 is less than 92), \( y \) increases to 91. So there is a negative trend. But the correlation coefficient is approximately -0.95, which is a strong negative correlation. But the options are "Weak" or "Strong". Since the absolute value of \( r \) is close to 1 (0.95), the relationship is strong.

Wait, maybe I miscalculated the sum of \( (x_i - \bar{x})(y_i - \bar{y}) \). Let's recalculate:

\( \bar{x} = 437 / 5 = 87.4 \)

\( \bar{y} = 436 / 5 = 87.2 \)

For \( (90, 80) \): \( (90 - 87.4) = 2.6 \), \( (80 - 87.2) = -7.2 \), product: \( 2.6 \times (-7.2) = -18.72 \)

For \( (87, 89) \): \( (87 - 87.4) = -0.4 \), \( (89 - 87.2) = 1.8 \), product: \( -0.4 \times 1.8 = -0.72 \)

For \( (83, 94) \): \( (83 - 87.4) = -4.4 \), \( (94 - 87.2) = 6.8 \), product: \( -4.4 \times 6.8 = -29.92 \)

For \( (92, 82) \): \( (92 - 87.4) = 4.6 \), \( (82 - 87.2) = -5.2 \), product: \( 4.6 \times (-5.2) = -23.92 \)

For \( (85, 91) \): \( (85 - 87.4) = -2.4 \), \( (91 - 87.2) = 3.8 \), product: \( -2.4 \times 3.8 = -9.12 \)

Summing these: \( -18.72 - 0.72 - 29.92 - 23.92 - 9.12 = -82.4 \) (correct)

Sum of \( (x_i - \bar{x})^2 \): \( 2.6^2 + (-0.4)^2 + (-4.4)^2 + 4.6^2 + (-2.4)^2 = 6.76 + 0.16 + 19.36 + 21.16 + 5.76 = 53.2 \) (correct)

Sum of \( (y_i - \bar{y})^2 \): \( (-7.2)^2 + 1.8^2 + 6.8^2 + (-5.2)^2 + 3.8^2 = 51.84 + 3.24 + 46.24 + 27.04 + 14.44 = 142.8 \) (correct)

So \( r = -82.4 / \sqrt{53.2 \times 142.8} = -82.4 / \sqrt{7596.96} = -82.4 / 87.16 \approx -0.945 \)

The absolute value is 0.945, which is close to 1, so the relationship is strong.

Answer:

Strong