Questions: Use the data set to do the following. a) Draw a scatter diagram. b) Determine the value of r. c) Determine whether a correlation exists at α=0.05. d) Determine whether a correlation exists at α=0.01. (i) Click the icon to view the correlation coefficient, r, for various values of n and α. x y --- --- 80 3 70 3 50 4 50 7 30 7 20 9 The formula for the correlation coefficient is r = (n(Σ xy) - (Σ x)(Σ y)) / sqrt(n(Σ x^2) - (Σ x)^2) sqrt(n(Σ y^2) - (Σ y)^2). a) Choose the scatter diagram that represents the given data. A. B. C.

Use the data set to do the following.
a) Draw a scatter diagram.
b) Determine the value of r.
c) Determine whether a correlation exists at α=0.05.
d) Determine whether a correlation exists at α=0.01.
(i) Click the icon to view the correlation coefficient, r, for various values of n and α.

x y
--- ---
80 3
70 3
50 4
50 7
30 7
20 9

The formula for the correlation coefficient is r = (n(Σ xy) - (Σ x)(Σ y)) / sqrt(n(Σ x^2) - (Σ x)^2) sqrt(n(Σ y^2) - (Σ y)^2).
a) Choose the scatter diagram that represents the given data.
A.
B.
C.

Transcript text: Use the data set to do the following. a) Draw a scatter diagram. b) Determine the value of $r$. c) Determine whether a correlation exists at $\alpha=0.05$. d) Determine whether a correlation exists at $\alpha=0.01$. (i) Click the icon to view the correlation coefficient, $r$, for various values of $n$ and $\alpha$. \begin{tabular}{c|c} $\boldsymbol{x}$ & $\mathbf{y}$ \\ \hline 80 & 3 \\ \hline 70 & 3 \\ \hline 50 & 4 \\ \hline 50 & 7 \\ \hline 30 & 7 \\ \hline 20 & 9 \end{tabular} The formula for the correlation coefficient is $\mathrm{r}=\frac{\mathrm{n}(\Sigma \mathrm{xy})-(\Sigma \mathrm{x})(\Sigma \mathrm{y})}{\sqrt{n\left(\Sigma x^{2}\right)-(\Sigma \mathrm{x})^{2}} \sqrt{n\left(\Sigma y^{2}\right)-(\Sigma y)^{2}}}$. a) Choose the scatter diagram that represents the given data. A. B. c.

Solution

Solution Steps

Step 1: Choose the scatter diagram that represents the given data

The data points are (80, 3), (70, 3), (50, 4), (50, 7), (30, 7), and (20, 9).
Plotting these points on the scatter diagrams provided, we see that the correct scatter diagram is B.

Step 2: Calculate the value of r

The formula for the correlation coefficient $ r $ is: \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} \]
Given data: \[ \begin{array}{|c|c|} \hline x & y \\ \hline 80 & 3 \\ 70 & 3 \\ 50 & 4 \\ 50 & 7 \\ 30 & 7 \\ 20 & 9 \\ \hline \end{array} \]
Calculate the necessary sums: \[ \sum x = 80 + 70 + 50 + 50 + 30 + 20 = 300 \] \[ \sum y = 3 + 3 + 4 + 7 + 7 + 9 = 33 \] \[ \sum xy = (80 \cdot 3) + (70 \cdot 3) + (50 \cdot 4) + (50 \cdot 7) + (30 \cdot 7) + (20 \cdot 9) = 240 + 210 + 200 + 350 + 210 + 180 = 1390 \] \[ \sum x^2 = 80^2 + 70^2 + 50^2 + 50^2 + 30^2 + 20^2 = 6400 + 4900 + 2500 + 2500 + 900 + 400 = 17600 \] \[ \sum y^2 = 3^2 + 3^2 + 4^2 + 7^2 + 7^2 + 9^2 = 9 + 9 + 16 + 49 + 49 + 81 = 213 \]
Plug these values into the formula: \[ r = \frac{6(1390) - (300)(33)}{\sqrt{[6(17600) - (300)^2][6(213) - (33)^2]}} \] \[ r = \frac{8340 - 9900}{\sqrt{[105600 - 90000][1278 - 1089]}} \] \[ r = \frac{-1560}{\sqrt{15600 \cdot 189}} \] \[ r = \frac{-1560}{\sqrt{2948400}} \] \[ r = \frac{-1560}{1717.67} \] \[ r \approx -0.91 \]

Step 3: Determine whether a correlation exists at $\alpha = 0.05$

For $ n = 6 $ and $\alpha = 0.05$, the critical value for $ r $ from the correlation table is approximately $ \pm 0.811 $.
Since $ |r| = 0.91 $ is greater than 0.811, we reject the null hypothesis and conclude that a significant correlation exists at $\alpha = 0.05$.