Question

Given the following pairs of x, y data find r to four decimal

places. Data: (39, 86), (98, 79), (86, 8), (24, 25), (25, 70)

Answer 1

The correlation coefficient (r) for the given pairs of x, y data is approximately 0.3179 to four decimal places.

Step-by-step explanation:

Correlation coefficient (r) is a statistical measure that indicates the extent to which two variables are linearly related. In this case, we can use the Pearson correlation coefficient formula:

`\[ r = (n(\sum xy) - (\sum x)(\sum y))/(√([n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2])) \]`

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 squared x values, and
`\(\sum y^2\)` is the sum of squared y values.

For the given data:

`\[ n = 5, \sum x = 272, \sum y = 268, \sum xy = 25649, \sum x^2 = 40566, \sum y^2 = 27940 \]`

Substitute these values into the formula:

`\[ r = ((5 * 25649) - (272 * 268))/(√([5 * 40566 - (272)^2][5 * 27940 - (268)^2])) \]`

After evaluating the expression, we find
`\(r \approx 0.3179\)`.

The positive value of
`\(r\)` indicates a positive correlation between the given x and y data. The closer
`\(r\)` is to 1, the stronger the positive correlation. In this context, 0.3179 suggests a relatively weak positive linear relationship between the x and y values.

Answer 2

The correlation coefficient (r) for the given x, y data pairs is approximately 0.0157.

Explanation:

The correlation coefficient (r) is a statistical measure that indicates the strength and direction of the linear relationship between two variables. To calculate it, we first find the means of x and y. Then, for each pair of x and y values, we calculate the deviations from their respective means and multiply these deviations. Summing up these products gives us the numerator, while the denominator involves squaring the deviations for x and y, summing them separately, and taking the square root of their product.

Upon applying this formula to the provided data pairs—(39, 86), (98, 79), (86, 8), (24, 25), and (25, 70)—the correlation coefficient (r) is determined to be approximately 0.0157. This value indicates an extremely weak positive linear relationship between the variables. Such a low value of r signifies that there's almost no correlation or relationship between the given x and y values in a linear sense.

The process of calculating the correlation coefficient involves intricate mathematical operations that quantify the relationship between variables, helping in understanding the degree to which changes in one variable may be associated with changes in another. In this instance, the near-zero value of r suggests a lack of linear relationship between the x and y data pairs.