Questions: Does the random paired data below show a linear correlation? x y ------ 2 16.44 3 28.56 4 21.18 5 27.8 6 25.62 7 27.54 8 35.76 9 34.28 10 33.8 11 32.42 12 37.74 13 34.66 14 44.78 15 38.4 When in doubt, assume the original claim is There is linear correlation.

Does the random paired data below show a linear correlation?

x y
------
2 16.44
3 28.56
4 21.18
5 27.8
6 25.62
7 27.54
8 35.76
9 34.28
10 33.8
11 32.42
12 37.74
13 34.66
14 44.78
15 38.4

When in doubt, assume the original claim is There is linear correlation.

Transcript text: Does the random paired data below show a linear correlation? \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 2 & 16.44 \\ \hline 3 & 28.56 \\ \hline 4 & 21.18 \\ \hline 5 & 27.8 \\ \hline 6 & 25.62 \\ \hline 7 & 27.54 \\ \hline 8 & 35.76 \\ \hline 9 & 34.28 \\ \hline 10 & 33.8 \\ \hline 11 & 32.42 \\ \hline 12 & 37.74 \\ \hline 13 & 34.66 \\ \hline 14 & 44.78 \\ \hline 15 & 38.4 \\ \hline \end{tabular} When in doubt, assume the original claim is There is linear correlation.

Solution

Solution Steps

Step 1: Calculate Covariance and Standard Deviations

The covariance between $ X $ and $ Y $ is calculated as: \[ \text{Cov}(X,Y) = 27.1577 \] The standard deviation of $ X $ is: \[ \sigma_X = 4.1833 \] The standard deviation of $ Y $ is: \[ \sigma_Y = 7.3929 \]

Step 2: Calculate Correlation Coefficient

The correlation coefficient $ r $ is computed using the formula: \[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} = 0.8781 \]

Step 3: Calculate Means of $ X $ and $ Y $

The mean of $ X $ is: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 8.5 \] The mean of $ Y $ is: \[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 31.3557 \]

Step 4: Calculate Slope $ \beta $

The numerator for $ \beta $ is calculated as: \[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 4084.38 - 14 \cdot 8.5 \cdot 31.3557 = 353.05 \] The denominator for $ \beta $ is: \[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 1239 - 14 \cdot 8.5^2 = 227.5 \] Thus, the slope $ \beta $ is: \[ \beta = \frac{353.05}{227.5} = 1.5519 \]

Step 5: Calculate Intercept $ \alpha $

The intercept $ \alpha $ is calculated using: \[ \alpha = \bar{y} - \beta \bar{x} = 31.3557 - 1.5519 \cdot 8.5 = 18.1648 \]

Step 6: Write the Equation of the Line of Best Fit

The equation of the line of best fit is: \[ y = 18.1648 + 1.5519x \]

Step 7: Summary of Results

The correlation coefficient from the linear regression is: \[ \text{Correlation Coefficient} = 0.8781 \] The intercept is: \[ \alpha = 18.1648 \] The slope is: \[ \beta = 1.5519 \]