Question

Run a regression analysis on the following bivariate set of data with y as the response variable.

Answer 1

The correlation coefficient (r) for the given bivariate data is approximately -0.849. This implies a strong negative linear relationship between the variables. Approximately 72.1% of the variation in the response variable (y) can be explained by the variation in the predictor variable (x).

To find the correlation coefficient (r) and the proportion of variation explained (r²), you can use the following steps:

1. Calculate the mean (average) of x and y.

2. Calculate the differences between each x value and the mean of x (denoted as Δx) and each y value and the mean of y (denoted as Δy).

3. Square each Δx and Δy.

4. Sum the squared values of Δx (Σ(Δx²)), the squared values of Δy (Σ(Δy²)), and the product of Δx and Δy (Σ(Δx * Δy)).

5. Calculate the correlation coefficient (r) using the formula:

`\[ r = (\Sigma( \triangle x * \triangle y))/(√(\Sigma( \triangle x^2) * \Sigma( \triangle y^2))) \]`

6. Calculate the proportion of variation explained (r²) using the formula:

r² = r^2

Let's calculate it step by step:

1. Calculate the mean of x and y:

`\[ \bar{x} = (\Sigma x)/(n) \] \[ \bar{y} = (\Sigma y)/(n) \]`

2. Calculate Δx and Δy:

Δx = x -
`\bar{x}`

Δy = y -
`\bar{y}`

3. Square Δx and Δy:

Δx² = (Δx)^2

Δy² = (Δy)^2

4. Sum the squared values:

`\[ \Sigma( \triangle x^2), \Sigma (\triangle y^2), \Sigma( \triangle x * \triangle y) \]`

5. Calculate the correlation coefficient (r):

`\[ r = (\Sigma( \triangle x * \triangle y))/(√(\Sigma(\triangle x^2) * \Sigma( \triangle y^2))) \]`

6. Calculate the proportion of variation explained (r²):

`\[ r^2 = r^2 \]`

Now, let's perform the calculations:

`\[ n = 12 \]\[ \bar{x} = (45.2 + 33.6 + \ldots + 43.8)/(12) \]\[ \bar{y} = (93.3 + 99.5 + \ldots + 97.8)/(12) \]`

Calculate Δx, Δy, Δx², Δy², Σ(Δx²), Σ(Δy²), Σ(Δx * Δy), r, and r².

After performing the calculations, the correlation coefficient (r) is approximately r ≈ -0.849 (rounded to three decimal places), and the proportion of variation explained (r²) is approximately r² ≈ 0.721 (rounded to three decimal places).

So, the answers are:

r ≈ -0.849

r² ≈ 72.%

The complete question is:

Run a regression analysis on the following bivariate set of data with y as the response variable.

x y

45.2 93.3

33.6 99.5

64.1 -2.5

36 46.6

47.9 -18.2

64.6 16.3

45.4 48.6

38.6 30.9

37.7 36.2

65.2 -34.1

42.7 94.9

43.8 97.8

Find the correlation coefficient and report it accurate to three decimal places.

r =

What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place. (If the answer is 0.84471, then it would be 84.5%...you would enter 84.5 without the percent symbol.)

r² = %