Determine The Linear Correlation Coefficient Between Commute Time And Well-being Score. Commute Time: 4, 13, 23, 36, 53, 75, 101. Well-being Score: 69.2, 68.9, 67.8, 67.7, 66.2, 65.8, 63.1. R=___?

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Introduction

In this article, we will explore the relationship between commute time and well-being score using the linear correlation coefficient. The linear correlation coefficient, denoted by r, measures the strength and direction of the linear relationship between two variables. In this case, we will examine the relationship between commute time and well-being score.

What is the Linear Correlation Coefficient?

The linear correlation coefficient, also known as Pearson's correlation coefficient, is a statistical measure that calculates the strength and direction of the linear relationship between two continuous variables. It is denoted by the symbol r and ranges from -1 to 1. A value of 1 indicates a perfect positive linear relationship, while a value of -1 indicates a perfect negative linear relationship. A value of 0 indicates no linear relationship between the variables.

Calculating the Linear Correlation Coefficient

To calculate the linear correlation coefficient, we need to follow these steps:

  1. Calculate the mean of both variables: The mean of the commute time and well-being score variables are calculated by summing up all the values and dividing by the number of values.

    • Commute time: (4 + 13 + 23 + 36 + 53 + 75 + 101) / 7 = 305 / 7 = 43.57
    • Well-being score: (69.2 + 68.9 + 67.8 + 67.7 + 66.2 + 65.8 + 63.1) / 7 = 478.7 / 7 = 68.43

  2. Calculate the deviations from the mean: The deviations from the mean are calculated by subtracting the mean from each value.

    • Commute time deviations: (4 - 43.57), (13 - 43.57), (23 - 43.57), (36 - 43.57), (53 - 43.57), (75 - 43.57), (101 - 43.57)
    • Well-being score deviations: (69.2 - 68.43), (68.9 - 68.43), (67.8 - 68.43), (67.7 - 68.43), (66.2 - 68.43), (65.8 - 68.43), (63.1 - 68.43)

  3. Calculate the covariance: The covariance is calculated by multiplying the deviations from the mean of both variables and summing them up.

    • Covariance = [(4 - 43.57)(69.2 - 68.43) + (13 - 43.57)(68.9 - 68.43) + (23 - 43.57)(67.8 - 68.43) + (36 - 43.57)(67.7 - 68.43) + (53 - 43.57)(66.2 - 68.43) + (75 - 43.57)(65.8 - 68.43) + (101 - 43.57)(63.1 - 68.43)]

  4. Calculate the variance: The variance is calculated by squaring the deviations from the mean and summing them up.

    • Variance of commute time = [(4 - 43.57)^2 + (13 - 43.57)^2 + (23 - 43.57)^2 + (36 - 43.57)^2 + (53 - 43.57)^2 + (75 - 43.57)^2 + (101 - 43.57)^2]
    • Variance of well-being score = [(69.2 - 68.43)^2 + (68.9 - 68.43)^2 + (67.8 - 68.43)^2 + (67.7 - 68.43)^2 + (66.2 - 68.43)^2 + (65.8 - 68.43)^2 + (63.1 - 68.43)^2]

  5. Calculate the linear correlation coefficient: The linear correlation coefficient is calculated by dividing the covariance by the square root of the product of the variances.

    • r = covariance / sqrt(variance of commute time * variance of well-being score)

Calculating the Linear Correlation Coefficient for the Given Data

Let's calculate the linear correlation coefficient for the given data.

Commute Time Well-being Score
4 69.2
13 68.9
23 67.8
36 67.7
53 66.2
75 65.8
101 63.1

Mean of Commute Time and Well-being Score

  • Mean of commute time = 305 / 7 = 43.57
  • Mean of well-being score = 478.7 / 7 = 68.43

Deviations from the Mean

  • Commute time deviations: (-39.57), (-30.57), (-20.57), (-7.57), (9.43), (31.43), (57.43)
  • Well-being score deviations: (0.77), (0.47), (-0.63), (-0.73), (-2.23), (-2.63), (-5.33)

Covariance

  • Covariance = [(-39.57)(0.77) + (-30.57)(0.47) + (-20.57)(-0.63) + (-7.57)(-0.73) + (9.43)(-2.23) + (31.43)(-2.63) + (57.43)(-5.33)]

Variance

  • Variance of commute time = [(-39.57)^2 + (-30.57)^2 + (-20.57)^2 + (-7.57)^2 + (9.43)^2 + (31.43)^2 + (57.43)^2]
  • Variance of well-being score = [(0.77)^2 + (0.47)^2 + (-0.63)^2 + (-0.73)^2 + (-2.23)^2 + (-2.63)^2 + (-5.33)^2]

Linear Correlation Coefficient

  • r = covariance / sqrt(variance of commute time * variance of well-being score)

Numerical Calculation

Let's perform the numerical calculation.

  • Covariance = (-30.59) + (14.33) + (13.01) + (5.53) + (-21.01) + (-82.51) + (-304.19) = -405.33
  • Variance of commute time = 1573.19 + 933.33 + 421.69 + 57.29 + 88.89 + 986.09 + 3293.29 = 8194.86
  • Variance of well-being score = 0.59 + 0.22 + 0.40 + 0.53 + 4.96 + 6.89 + 28.37 = 42.96
  • r = -405.33 / sqrt(8194.86 * 42.96) = -405.33 / 143.19 = -2.83

Conclusion

In this article, we calculated the linear correlation coefficient between commute time and well-being score using the given data. The linear correlation coefficient, denoted by r, measures the strength and direction of the linear relationship between two variables. In this case, we found that the linear correlation coefficient is -2.83, indicating a strong negative linear relationship between commute time and well-being score.

Limitations

This analysis has several limitations. Firstly, the sample size is small, which may not be representative of the population. Secondly, the data may not be normally distributed, which may affect the accuracy of the results. Finally, the analysis assumes a linear relationship between the variables, which may not be the case in reality.

Future Research Directions

Future research directions may include:

  • Increasing the sample size: To increase the sample size and make the results more representative of the population.
  • Checking for normality: To check if the data is normally distributed and adjust the analysis accordingly.
  • Examining non-linear relationships: To examine if there are any non-linear relationships between the variables.

Q: What is the linear correlation coefficient?

A: The linear correlation coefficient, denoted by r, measures the strength and direction of the linear relationship between two continuous variables. It ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship.

Q: How is the linear correlation coefficient calculated?

A: The linear correlation coefficient is calculated using the following steps:

  1. Calculate the mean of both variables.
  2. Calculate the deviations from the mean for both variables.
  3. Calculate the covariance by multiplying the deviations from the mean of both variables and summing them up.
  4. Calculate the variance of both variables by squaring the deviations from the mean and summing them up.
  5. Calculate the linear correlation coefficient by dividing the covariance by the square root of the product of the variances.

Q: What is the difference between correlation and causation?

A: Correlation and causation are two related but distinct concepts. Correlation refers to the relationship between two variables, while causation refers to the cause-and-effect relationship between two variables. Just because two variables are correlated, it does not mean that one variable causes the other.

Q: What is the significance of the linear correlation coefficient?

A: The linear correlation coefficient is significant because it helps to:

  1. Identify relationships: The linear correlation coefficient helps to identify the strength and direction of the linear relationship between two variables.
  2. Make predictions: The linear correlation coefficient can be used to make predictions about the value of one variable based on the value of the other variable.
  3. Analyze data: The linear correlation coefficient is a useful tool for analyzing data and identifying patterns and trends.

Q: What are the limitations of the linear correlation coefficient?

A: The linear correlation coefficient has several limitations, including:

  1. Assumes linearity: The linear correlation coefficient assumes a linear relationship between the variables, which may not be the case in reality.
  2. Sensitive to outliers: The linear correlation coefficient is sensitive to outliers, which can affect the accuracy of the results.
  3. Does not account for non-linear relationships: The linear correlation coefficient does not account for non-linear relationships between the variables.

Q: How can I interpret the linear correlation coefficient?

A: The linear correlation coefficient can be interpreted as follows:

  1. Positive correlation: A positive correlation indicates that as one variable increases, the other variable also increases.
  2. Negative correlation: A negative correlation indicates that as one variable increases, the other variable decreases.
  3. Zero correlation: A zero correlation indicates that there is no linear relationship between the variables.

Q: What are some common applications of the linear correlation coefficient?

A: The linear correlation coefficient has several common applications, including:

  1. Economics: The linear correlation coefficient is used to analyze the relationship between economic variables, such as GDP and inflation.
  2. Finance: The linear correlation coefficient is used to analyze the relationship between financial variables, such as stock prices and interest rates.
  3. Social sciences: The linear correlation coefficient is used to analyze the relationship between social variables, such as education and income.

Q: How can I calculate the linear correlation coefficient using a calculator or computer software?

A: The linear correlation coefficient can be calculated using a calculator or computer software, such as Excel or R. The steps are as follows:

  1. Enter the data into the calculator or computer software.
  2. Use the built-in function to calculate the linear correlation coefficient.
  3. Interpret the results and draw conclusions based on the analysis.

By following these steps and understanding the limitations and applications of the linear correlation coefficient, you can use this powerful tool to analyze data and make informed decisions.