To address the questions:

(a) Regression Line of Time Taken on Month’s Experience

The regression line is given by the formula: $y = a + bx$ Where:

• $y$: Dependent variable (time taken)
• $x$: Independent variable (month's experience)
• $b$: Slope of the regression line $b = \frac{\sum(xy) - n\bar{x}\bar{y}}{\sum(x^2) - n\bar{x}^2}$
• $a$: Intercept of the regression line $a = \bar{y} - b\bar{x}$

(b) Coefficient of Correlation ($r$) and Coefficient of Determination ($r^2$)

The formula for the correlation coefficient ($r$) is: [ r = \frac{\sum(xy) - n\bar{x}\bar{y}}{\sqrt{[\sum(x^2) - n\bar{x}^2][\sum(y^2) - n\bar{y}^2]}} ] The coefficient of determination ($r^2$) measures the proportion of variance in the dependent variable explained by the independent variable.

Let’s compute these using the given data:

Month’s Experience ($x$)	Time Taken ($y$)
2	27
5	26
3	30
8	20
5	22
9	20
12	16
16	15
1	30
6	19

I will calculate the required values.import pandas as pd import numpy as np

Data

data = { "Months_Experience": [2, 5, 3, 8, 5, 9, 12, 16, 1, 6], "Time_Taken": [27, 26, 30, 20, 22, 20, 16, 15, 30, 19] } df = pd.DataFrame(data)

Calculations

x = df["Months_Experience"] y = df["Time_Taken"] n = len(df)

Summations

sum_x = np.sum(x) sum_y = np.sum(y) sum_x2 = np.sum(x2) sum_y2 = np.sum(y2) sum_xy = np.sum(x * y)

Means

mean_x = np.mean(x) mean_y = np.mean(y)

Regression coefficients

b = (sum_xy - n * mean_x * mean_y) / (sum_x2 - n * mean_x**2) a = mean_y - b * mean_x

Correlation coefficient (r)

r = (sum_xy - n * mean_x * mean_y) / np.sqrt((sum_x2 - n * mean_x2) * (sum_y2 - n * mean_y2))

Coefficient of determination (r^2)

r_squared = r**2

Results

regression_line = f"y = {a:.2f} + {b:.2f}x" regression_line, r, r_squared('y = 29.59 + -1.06x', -0.904288815081655, 0.8177382610817837)('y = 29.59 + -1.06x', -0.904288815081655, 0.8177382610817837)success### Results:

(a) Regression Line:

The regression line of time taken ($y$) on months of experience ($x$) is: $y = 29.59 - 1.06x$

(b) Coefficients:

• Correlation Coefficient ($r$): $-0.904$
  This indicates a strong negative correlation between time taken and months of experience; as experience increases, time taken decreases significantly.

• Coefficient of Determination ($r^2$): $0.818$
  Approximately $81.8\%$ of the variance in the time taken is explained by the worker's months of experience.

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Would you like a detailed step-by-step breakdown of the calculations? Let me know if you have questions or need further clarification.

Related Questions:

1. How does a steep learning curve affect workers' output over time?
2. Can this regression model predict performance for new employees with zero experience?
3. What are some ways to mitigate discrimination in a results-based pay system?
4. Would a nonlinear model better fit this data, given the learning curve?
5. How can the company use this data to improve the payment system?

Tip:

Always analyze the residuals of your regression to verify if the linear model is appropriate for your data.