It seems you have provided a dataset representing the output (Y) produced and labor input (X) used for a hypothetical firm over 10 observations. Let's see how we can analyze this data.
Here's the provided dataset summarized:
| Obs. (Firm) | Y | X | |--------------|----|---| | 1 | 11 | 10| | 2 | 10 | 7 | | 3 | 12 | 10| | 4 | 6 | 5 | | 5 | 10 | 8 | | 6 | 7 | 8 | | 7 | 9 | 6 | | 8 | 10 | 7 | | 9 | 11 | 9 | | 10 | 10 | 10|
Possible Analyses
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Calculate Average Output and Labor Input:
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Average Output (Y): \[ \text{Average } Y = \frac{11 + 10 + 12 + 6 + 10 + 7 + 9 + 10 + 11 + 10}{10} = \frac{ 10 + 12 + 6 + 10 + 7 + 9 + 10 + 11 +10 }{10} = 9.9 \]
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Average Labor Input (X): \[ \text{Average } X = \frac{10 + 7 + 10 + 5 + 8 + 8 + 6 + 7 + 9 + 10}{10} = \frac{ 10 + 7 + 10 + 5 + 8 + 8 + 6 + 7 + 9 + 10 }{10} = 8.0 \]
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Calculate the Correlation Between Y and X: To calculate the correlation coefficient between Y and X, we would typically use the Pearson correlation formula: \[ r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}} \]
First, we need the sums of Y, X, XY, \(X^2\), and \(Y^2\):
- \(\sum Y = 106\)
- \(\sum X = 79\)
- \(\sum XY = 1110 + 107 + 1210 + 65 + 108 + 78 + 96 + 107 + 119 + 1010 = 110 + 70 + 120 + 30 + 80 + 56 + 54 + 70 + 99 + 100 = 709\)
- \(\sum X^2 = 10^2 + 7^2 + 10^2 + 5^2 + 8^2 + 8^2 + 6^2 + 7^2 + 9^2 + 10^2 = 100 + 49 + 100 + 25 + 64 + 64 + 36 + 49 + 81 + 100 = 568\)
- \(\sum Y^2 = 11^2 + 10^2 + 12^2 + 6^2 + 10^2 + 7^2 + 9^2 + 10^2 + 11^2 + 10^2 = 121 + 100 + 144 + 36 + 100 + 49 + 81 + 100 + 121 + 100 = 962\)
Now plug in these values into the correlation formula: \[ r = \frac{10(709) - (79)(106)}{\sqrt{[10(568) - (79)^2][10(962) - (106)^2]}} = \frac{7090 - 8374}{\sqrt{[5680 - 6241][9620 - 11236]}} = \frac{-1284}{\sqrt{[-561][-1606]}} \] This result will yield a correlation coefficient indicating how labor input relates to output.
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Regression Analysis: You might also want to perform a simple linear regression analysis to determine the relationship between Y (output) and X (labor input). The regression equation typically takes the form: \[ Y = \beta_0 + \beta_1 X + \epsilon \] Where \(\beta_0\) is the intercept, \(\beta_1\) is the slope (the effect of X on Y), and \(\epsilon\) is the error term.
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Graphical Representation: It could also be helpful to create a scatter plot of the data to visualize the relationship between labor input (X) and output (Y).
If you need further analyses or specific computations based on this data, feel free to specify!