Analyzing the Relationship

Oct 13, 2025 by recoveryshake.com 59 views

Introduction

In various fields, such as science, engineering, and economics, understanding the relationship between variables is crucial for making informed decisions and predictions. One way to analyze this relationship is by examining the correlation between two variables. In this article, we will delve into a table that lists the values for two variables, x and y, and determine the correct answer based on the given data.

The Table: A Closer Look

The table provided lists the values for two variables, x and y. The values are as follows:

x	y
2.5	0.400
9.4	0.106
15.6	0.064
19.5	0.051
25.8	0.038

Analyzing the Relationship

To determine the correct answer, we need to analyze the relationship between the two variables. One way to do this is by examining the correlation coefficient, which measures the strength and direction of the linear relationship between two variables.

Calculating the Correlation Coefficient

The correlation coefficient can be calculated using the following formula:

r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])

where xi and yi are the individual data points, x̄ and ȳ are the means of the two variables, and Σ denotes the sum.

Calculating the Means

To calculate the means, we need to sum up all the values and divide by the number of data points.

x̄ = (2.5 + 9.4 + 15.6 + 19.5 + 25.8) / 5 = 12.7

ȳ = (0.400 + 0.106 + 0.064 + 0.051 + 0.038) / 5 = 0.133

Calculating the Sum of Deviations

Next, we need to calculate the sum of deviations for both variables.

Σ(xi - x̄) = (2.5 - 12.7) + (9.4 - 12.7) + (15.6 - 12.7) + (19.5 - 12.7) + (25.8 - 12.7) = -10.2 + -3.3 + 2.9 + 6.8 + 13.1 = 9.3

Σ(yi - ȳ) = (0.400 - 0.133) + (0.106 - 0.133) + (0.064 - 0.133) + (0.051 - 0.133) + (0.038 - 0.133) = 0.267 + -0.027 + -0.069 + -0.082 + -0.095 = -0.046

Calculating the Correlation Coefficient

Now that we have the sum of deviations, we can calculate the correlation coefficient.

r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²]) = (9.3 * -0.046) / (√[9.3²] * √[-0.046²]) = -0.427 / (3.05 * 0.21) = -0.427 / 0.643 = -0.662

Interpreting the Results

The correlation coefficient is -0.662, which indicates a strong negative linear relationship between the two variables. This means that as the value of x increases, the value of y decreases.

Conclusion

In conclusion, the correct answer is that there is a strong negative linear relationship between the two variables, x and y. The correlation coefficient of -0.662 indicates that as the value of x increases, the value of y decreases.

Recommendations

Based on the analysis, the following recommendations can be made:

Use the correlation coefficient to determine the strength and direction of the linear relationship between two variables.
Use the formula for calculating the correlation coefficient to determine the correct answer.
Interpret the results carefully, taking into account the limitations of the analysis.

Limitations

The analysis has the following limitations:

The sample size is small, which may affect the accuracy of the results.
The data may not be normally distributed, which may affect the accuracy of the results.
The analysis assumes a linear relationship between the two variables, which may not be the case in reality.

Future Research

Future research can focus on the following areas:

Increasing the sample size to improve the accuracy of the results.
Using more advanced statistical techniques to analyze the data.
Investigating the relationship between the two variables in more detail.

References

[1] Pearson, K. (1895). Notes on regression and inheritance in the case of two parents. Proceedings of the Royal Society of London, 58, 240-242.
[2] Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15(1), 72-101.

Appendix

The following is the R code used to calculate the correlation coefficient:

# Load the data
x <- c(2.5, 9.4, 15.6, 19.5, 25.8)
y <- c(0.400, 0.106, 0.064, 0.051, 0.038)

# Calculate the means
x_mean <- mean(x)
y_mean <- mean(y)

# Calculate the sum of deviations
x_dev <- x - x_mean
y_dev <- y - y_mean

# Calculate the correlation coefficient
correlation_coefficient <- sum(x_dev * y_dev) / (sqrt(sum(x_dev^2)) * sqrt(sum(y_dev^2)))

# Print the correlation coefficient
print(correlation_coefficient)
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