Questions: Correlation and Simple Linear Regression Performing a simple linear regression Calf age, x (in months) Calf weight, y (in kg) Table Sample x y xy x^2 y^2 1 65 65 1 4225 3 110 330 9 12100 5 150 750 25 22500 7 178 1246 49 31684 9 244 2196 81 59536 11 292 3212 121 85264 Σ 36 1039 7799 286 215309 Based on the data from your sample, enter the indicated values in the column on the left (round decimal values to three decimal places). Note that n is the sample size and the symbol Σxy means the sum of the values xy. n: □ Σx: □ Σy: □ Σxy: □ Σx^2: □ Σy^2: □ Sample correlation coefficient (r): □ Slope (b): □ Intercept (a): □

Correlation and Simple Linear Regression Performing a simple linear regression Calf age, x (in months) Calf weight, y (in kg) Table Sample x y xy x^2 y^2 1 65 65 1 4225 3 110 330 9 12100 5 150 750 25 22500 7 178 1246 49 31684 9 244 2196 81 59536 11 292 3212 121 85264 Σ 36 1039 7799 286 215309 Based on the data from your sample, enter the indicated values in the column on the left (round decimal values to three decimal places). Note that n is the sample size and the symbol Σxy means the sum of the values xy. n: □ Σx: □ Σy: □ Σxy: □ Σx^2: □ Σy^2: □ Sample correlation coefficient (r): □ Slope (b): □ Intercept (a): □
Transcript text: Correlation and Simple Linear Regression Performing a simple linear regression Calf age, x (in months) Calf weight, y (in kg) Table Sample x y xy x^2 y^2 1 65 65 1 4225 3 110 330 9 12100 5 150 750 25 22500 7 178 1246 49 31684 9 244 2196 81 59536 11 292 3212 121 85264 Σ 36 1039 7799 286 215309 Based on the data from your sample, enter the indicated values in the column on the left (round decimal values to three decimal places). Note that n is the sample size and the symbol Σxy means the sum of the values xy. n: □ Σx: □ Σy: □ Σxy: □ Σx^2: □ Σy^2: □ Sample correlation coefficient (r): □ Slope (b): □ Intercept (a): □
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Solution

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Solution Steps

Step 1: Calculate Summations

We have the following data for calf age \( x \) (in months) and calf weight \( y \) (in kg):

\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 65 \\ 3 & 110 \\ 5 & 150 \\ 7 & 178 \\ 9 & 244 \\ 11 & 292 \\ \hline \end{array} \]

From this data, we calculate the following summations:

  • Sample size \( n = 6 \)
  • \( \Sigma x = 36 \)
  • \( \Sigma y = 1039 \)
  • \( \Sigma xy = 7799 \)
  • \( \Sigma x^2 = 286 \)
  • \( \Sigma y^2 = 215309 \)
Step 2: Calculate Correlation Coefficient

The formula to calculate the correlation coefficient \( r \) is:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

Where:

  • Covariance \( \text{Cov}(X,Y) = 313.0 \)
  • Standard deviation of \( X \) is \( \sigma_X = 3.742 \)
  • Standard deviation of \( Y \) is \( \sigma_Y = 84.129 \)

Thus, we find:

\[ r = \frac{313.0}{3.742 \times 84.129} = 0.994 \]

Step 3: Calculate Slope and Intercept

To find the slope \( \beta \) and intercept \( \alpha \) of the regression line, we use the following formulas:

  1. Slope \( \beta \):

\[ \beta = \frac{\sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y}}{\sum_{i=1}^{n} x_i^2 - n \bar{x}^2} \]

Where:

  • \( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 6.0 \)
  • \( \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 173.167 \)

Calculating the numerator:

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 7799 - 6 \times 6.0 \times 173.167 = 1565.0 \]

Calculating the denominator:

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 286 - 6 \times 6.0^2 = 70.0 \]

Thus, the slope is:

\[ \beta = \frac{1565.0}{70.0} = 22.357 \]

  1. Intercept \( \alpha \):

\[ \alpha = \bar{y} - \beta \bar{x} = 173.167 - 22.357 \times 6.0 = 39.024 \]

Final Answer

The results are summarized as follows:

  • Sample size \( n = 6 \)
  • \( \Sigma x = 36 \)
  • \( \Sigma y = 1039 \)
  • \( \Sigma xy = 7799 \)
  • \( \Sigma x^2 = 286 \)
  • \( \Sigma y^2 = 215309 \)
  • Sample correlation coefficient \( r = 0.994 \)
  • Slope \( b = 22.357 \)
  • Intercept \( a = 39.024 \)

The equation of the line of best fit is:

\[ y = 39.024 + 22.357x \]

Thus, the final boxed answers are:

\[ \boxed{n = 6} \] \[ \boxed{\Sigma x = 36} \] \[ \boxed{\Sigma y = 1039} \] \[ \boxed{\Sigma xy = 7799} \] \[ \boxed{\Sigma x^2 = 286} \] \[ \boxed{\Sigma y^2 = 215309} \] \[ \boxed{r = 0.994} \] \[ \boxed{b = 22.357} \] \[ \boxed{a = 39.024} \]

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