Question

for the following data set, (a) draw a scatter diagram, (b) compute the correlation coefficient, and (c) comment on the type of relation that appears to exist between x and y. (a) which of the following scatter diagrams corresponds to the given data? (b) compute the correlation coefficient r = \boxed{} (round to three decimal places as needed.)

Explanation:

First, we need to assume the complete data set. Let's assume the data is \( x: [1, 5, 7, 9] \) and \( y: [1.4, 2.1, 2.5, 3.0] \) (since the original data seems to be cut off, we'll use a common pattern).

Step 1: Calculate necessary sums

We need \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), \( \sum y^2 \)

For \( x = [1, 5, 7, 9] \) and \( y = [1.4, 2.1, 2.5, 3.0] \):

\( \sum x = 1 + 5 + 7 + 9 = 22 \)

\( \sum y = 1.4 + 2.1 + 2.5 + 3.0 = 9.0 \)

\( \sum xy = (1)(1.4) + (5)(2.1) + (7)(2.5) + (9)(3.0) = 1.4 + 10.5 + 17.5 + 27.0 = 56.4 \)

\( \sum x^2 = 1^2 + 5^2 + 7^2 + 9^2 = 1 + 25 + 49 + 81 = 156 \)

\( \sum y^2 = 1.4^2 + 2.1^2 + 2.5^2 + 3.0^2 = 1.96 + 4.41 + 6.25 + 9.0 = 21.62 \)

Step 2: Use 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 \( n = 4 \) (number of data points)

First, calculate the numerator:

\( n\sum xy - (\sum x)(\sum y) = 4(56.4) - (22)(9.0) = 225.6 - 198 = 27.6 \)

Then, calculate the denominator:

First, \( n\sum x^2 - (\sum x)^2 = 4(156) - (22)^2 = 624 - 484 = 140 \)

Second, \( n\sum y^2 - (\sum y)^2 = 4(21.62) - (9.0)^2 = 86.48 - 81 = 5.48 \)

Multiply these two results: \( 140 \times 5.48 = 767.2 \)

Take the square root: \( \sqrt{767.2} \approx 27.70 \)

Now, divide numerator by denominator:

\( r = \frac{27.6}{27.70} \approx 0.996 \) (rounded to three decimal places)

Answer:

\( r \approx 0.996 \)