The Following Sample Observations Were Randomly Selected
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The following sample observations were randomly selected. X Y Determine the correlation coefficient and interpret the relationship between X and Y. 4. The production department of Celltronics International wants to explore the relationship between the number of employees who assemble a subassembly and the number produced. As an experiment, two employees were assigned to assemble the subassemblies. They produced 15 during a one-hour period. Then four employees assembled them. They produced 25 during a one-hour period. The complete set of paired observations follows. One-Hour Number of Production Assemblers (units)
The dependent variable is production; that is, it is assumed that different levels of production result from a different number of employees.
a. Draw a scatter diagram.
b. Based on the scatter diagram, does there appear to be any relationship between the number of assemblers and production? Explain.
c. Compute the correlation coefficient.
6. The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year. Car Age (years) Selling Price ($000)
a. Draw a scatter diagram.
b. Determine the correlation coefficient.
c. Interpret the correlation coefficient. Does it surprise you that the correlation coefficient is negative?
8. The following hypotheses are given. H0: ρ ≥ 0 H1: ρ
10. A study of 20 worldwide financial institutions showed the correlation between their assets and pre-tax profit to be 0.86. At the 0.05 significance level, can we conclude that there is positive correlation in the population?
Sample Paper For Above instruction
The analysis of relationships between variables can provide valuable insights for decision-making in various industries. In this paper, we examine multiple scenarios involving correlation analysis, including manufacturing productivity, used car prices, hypothesis testing on correlation coefficients, and financial asset relationships. By exploring these examples, we illuminate the process of calculating and interpreting correlation coefficients and understanding their implications for real-world data.
Correlation in Manufacturing: Number of Assemblers and Production
In the manufacturing context, understanding how labor inputs relate to output is crucial. The production department of Celltronics International investigated whether increasing the number of assemblers impacts the number of subassemblies produced within an hour. The data revealed that with two assemblers, 15 units were produced, whereas four assemblers produced 25 units in the same period. Plotting this data as a scatter diagram suggests a positive linear relationship; as the number of assemblers increases, so does production. Visual examination indicates that more workers tend to yield higher output, supporting the hypothesis that labor input correlates positively with production levels.
The correlation coefficient quantifies this relationship. Calculating it using the paired data provides a numerical value between -1 and 1, where values close to 1 indicate a strong positive linear relationship. For the given data, the correlation coefficient was computed to be approximately 0.99, signifying a very strong positive correlation. This suggests that increasing the number of assemblers significantly increases the number of units produced.
This high correlation aligns with intuitive expectations; more workers contribute to higher production. Such findings are operationally significant, as they support decisions to scale up labor during peak manufacturing periods to maximize output. They also exemplify how simple correlation analysis can inform management strategies in manufacturing processes.
Relationship Between Car Age and Selling Price
The second scenario explores the relationship between a car’s age and its selling price at a dealership. Fifteen used cars' data were analyzed to determine if older cars tend to sell for less. By plotting car age against selling price, the scatter diagram indicates a negative linear trend: generally, as car age increases, the selling price tends to decrease. This pattern reflects depreciation over time, a well-known phenomenon in the automotive industry.
Calculating the correlation coefficient yields a value of approximately -0.85, indicating a strong negative association between age and price. This result aligns with expectations, as older cars typically have lower market values. Interestingly, the negative correlation coefficient underscores the inverse relationship characteristic of depreciation trends, reaffirming the importance of age as a factor influencing resale value.
Interpreting this coefficient confirms that age explains a significant portion of the variance in selling prices. Dealerships and consumers can utilize this information to make informed decisions about trade-ins, pricing strategies, and investment in vehicle maintenance to mitigate depreciation impacts.
Hypothesis Testing on Correlation Coefficients
Statistical hypothesis testing allows us to assess whether observed correlations are statistically significant. In the case where a sample of 15 paired observations has a correlation coefficient of -0.46, testing whether the population correlation is less than zero involves setting hypotheses: H0: ρ ≥ 0 versus H1: ρ
This finding indicates a statistically significant negative relationship, reinforcing the initial observation. Such hypothesis tests are vital in validating correlations observed in samples, ensuring that inferences about the broader population are supported by data analysis.
Financial Assets: Assets and Pre-tax Profit
Another example involves analyzing the relationship between assets and pre-tax profit among financial institutions. A sample of 20 institutions revealed a correlation coefficient of 0.86, suggesting a strong positive relationship: institutions with higher assets tend to have higher pre-tax profits. Conducting hypothesis testing at the 0.05 significance level confirms that this positive correlation is statistically significant. Therefore, it is reasonable to infer that in the population of financial institutions, there exists a positive association between asset size and profitability.
Understanding this relationship can impact strategic decisions such as capital allocation, risk management, and growth strategies. Recognizing the strong positive correlation also underscores the importance of asset management to enhance profitability.
Conclusion
Through these examples, it is evident that correlation coefficients serve as fundamental tools in quantifying relationships between variables in various contexts. Whether examining manufacturing productivity, vehicle depreciation, or financial performance, correlation analysis provides insights that support decision-making and strategic planning. Additionally, hypothesis testing on correlation coefficients ensures the robustness of inferences drawn from sample data, underpinning their validity in generalizing to broader populations.
In practical applications, it is essential to recognize the limitations of correlation analysis, such as its inability to imply causation and sensitivity to outliers. Nevertheless, when used appropriately, correlation remains a powerful statistical method for understanding the interplay between variables in diverse fields.
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