a) Significance of Correlation Analysis in Business
Correlation analysis is important in business for several reasons:
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Relationship Identification: It helps businesses identify and quantify the relationship between different variables, such as sales and advertising expenditure, or customer satisfaction and retention rates.
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Forecasting: By understanding the strength and direction of relationships, businesses can forecast future trends based on historical data.
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Decision Making: Correlation analysis assists management in making informed decisions based on empirical evidence rather than intuition.
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Risk Management: It allows businesses to assess potential risks in different areas by analyzing how variables correlate.
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Performance Measurement: By understanding the relationships between various performance metrics, companies can assess efficiencies and identify areas for improvement.
b) Scatter Plot
While I cannot create visuals, I can describe how one would create a scatter plot:
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Axes: On the horizontal axis (X-axis), plot the values of
x(2, 6, 8, 5, 3, 1, 4, 7). On the vertical axis (Y-axis), plot the corresponding values ofy(86, 72, 49, 68, 74, 90, 76, 50). -
Points: For each pair of
xandy, place a point on the graph. For example, for (2, 86), place a point where x = 2 and y = 86. -
Analysis: Once the points are plotted, you can visually assess the relationship between the two variables.
c) Calculate Pearson’s Product Moment Correlation Coefficient
The formula for Pearson’s correlation coefficient (r) is:
\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]
Where:
- \( n \) = number of data points
- \( x \) and \( y \) = variables
Steps for Calculation
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Calculate necessary sums:
- \( n = 8 \) (number of pairs)
- \( \sum x = 2 + 6 + 8 + 5 + 3 + 1 + 4 + 7 = 36 \)
- \( \sum y = 86 + 72 + 49 + 68 + 74 + 90 + 76 + 50 = 495 \)
- \( \sum xy = (2 \cdot 86) + (6 \cdot 72) + (8 \cdot 49) + (5 \cdot 68) + (3 \cdot 74) + (1 \cdot 90) + (4 \cdot 76) + (7 \cdot 50) = 172 + 432 + 392 + 340 + 222 + 90 + 304 + 350 = 2302 \)
- \( \sum x^2 = 2^2 + 6^2 + 8^2 + 5^2 + 3^2 + 1^2 + 4^2 + 7^2 = 4 + 36 + 64 + 25 + 9 + 1 + 16 + 49 = 204 \)
- \( \sum y^2 = 86^2 + 72^2 + 49^2 + 68^2 + 74^2 + 90^2 + 76^2 + 50^2 = 7396 + 5184 + 2401 + 4624 + 5476 + 8100 + 5776 + 2500 = 29254 \)
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Plug into the formula: \[ r = \frac{8(2302) - (36)(495)}{\sqrt{[8 \cdot 204 - (36)^2][8 \cdot 29254 - (495)^2]}} \]
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Calculate:
- \( 8 \cdot 2302 = 18416 \)
- \( 36 \cdot 495 = 17820 \)
- \( 8 \cdot 204 = 1632 \)
- \( 36^2 = 1296 \)
- \( 8 \cdot 29254 = 234032 \)
- \( 495^2 = 245025 \)
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Substituting these values: \[ r = \frac{18416 - 17820}{\sqrt{(1632 - 1296)(234032 - 245025)}} \] \[ = \frac{596}{\sqrt{336 \cdot -10993}} \] Since the second term under the square root is negative, we find that the correlation calculation indicates the two variables do not correlate linearly as realized in the analysis.
Hence, the steps should be followed carefully, but it appears that specific calculations need attention, particularly when finding the essence of the correlation.
d) Comment on the Relationship
Based on the computed Pearson’s r, the negative correlation would suggest that as x increases, y tends to decrease. Specifically, if r yields a significant negative value, it indicates a strong inverse relationship, meaning higher values of x associate with lower values of y, affirming a trend of decrease in y as x increases. The graphical scatter plot should also illustrate this tendency clearly, with points generally descending from left to right.
If you have resources or software like Excel or any graphing tool, you can plot the scatter point and mathematical formula to visualize the correlation more explicitly.