https://file.aiquickdraw.com/mathbot/file/1733239125553zptvbrx6.jpg

To answer the questions provided in the assignment, we will break down the steps systematically:

---

### **Part (a): Model Representation**
The model assumes a linear relationship between the independent variable \(x\) (Panjang, in meters) and the dependent variable \(y\) (Arus, in mA). The deterministic component is defined as:

\[
E(y) = \beta_0 + \beta_1 x
\]

To find \(\beta_0\) (intercept) and \(\beta_1\) (slope), we use the formulas:

1. **Slope (\(\beta_1\)):**
\[
\beta_1 = \frac{n\sum(xy) - \sum(x)\sum(y)}{n\sum(x^2) - (\sum(x))^2}
\]

2. **Intercept (\(\beta_0\)):**
\[
\beta_0 = \bar{y} - \beta_1 \bar{x}
\]
Where:
- \(n\) = number of data points
- \(x\) = Panjang
- \(y\) = Arus
- \(\bar{x}\) and \(\bar{y}\) are the means of \(x\) and \(y\), respectively.

---

### **Part (b): Sum of Squared Errors (SSE)**
SSE is calculated as:
\[
SSE = \sum (y_i - \hat{y}_i)^2
\]
Where:
- \(y_i\) = actual value
- \(\hat{y}_i = \beta_0 + \beta_1 x_i\) = predicted value

---

### **Steps to Calculate**
Let’s compute the necessary components for the calculations based on the data provided:

| Panjang (\(x\)) | Arus (\(y\)) | \(x^2\) | \(xy\) |
|------------------|--------------|---------|-------|
| 1               | 4            | 1       | 4     |
| 2               | 3            | 4       | 6     |
| 3               | 2            | 9       | 6     |
| 4               | 2            | 16      | 8     |
| 5               | 1            | 25      | 5     |

Let’s calculate the following:
- \(\sum x\), \(\sum y\), \(\sum x^2\), and \(\sum xy\)
- \(\beta_0\) and \(\beta_1\)
- SSE

I will now calculate these values. pandas as pd

# Data
data = {'Panjang (x)': [1, 2, 3, 4, 5], 
        'Arus (y)': [4, 3, 2, 2, 1]}
df = pd.DataFrame(data)

# Calculate required sums
df['x^2'] = df['Panjang (x)'] ** 2
df['xy'] = df['Panjang (x)'] * df['Arus (y)']

# Summing the values
sum_x = df['Panjang (x)'].sum()
sum_y = df['Arus (y)'].sum()
sum_x2 = df['x^2'].sum()
sum_xy = df['xy'].sum()
n = len(df)

# Calculate beta1 and beta0
beta1 = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x ** 2)
beta0 = (sum_y / n) - beta1 * (sum_x / n)

# Predicted y values and SSE
df['y_pred'] = beta0 + beta1 * df['Panjang (x)']
df['(y - y_pred)^2'] = (df['Arus (y)'] - df['y_pred']) ** 2
sse = df['(y - y_pred)^2'].sum()

# Display results
import ace_tools as tools; tools.display_dataframe_to_user(name="Regression and SSE Calculation Results", dataframe=df)

(beta0, beta1, sse)The calculations have been completed, and the table with detailed regression analysis results has been displayed for review. Here are the key findings:

1. **Intercept (\(\beta_0\)):** 4.5
2. **Slope (\(\beta_1\)):** -0.7
3. **Minimum Sum of Squared Error (SSE):** 0.3

Would you like further explanation on the steps or additional calculations?

Here are 5 related questions for deeper exploration:
1. How is the slope \(\beta_1\) affected by changes in the dataset?
2. Can we predict the current (\(y\)) for a new length (\(x\)) using this regression equation?
3. What does the low SSE indicate about the fit of the model?
4. How would the results change if an outlier were introduced into the data?
5. What are alternative ways to evaluate the goodness of fit for this regression model?

**Tip:** Always plot your regression line against the data points to visually assess the model's fit.

Dilan is studying the relationship between the length of a conductor and the electric current passing through it. Given the data on length (in meters) and current (in mA), the task involves determining the linear regression model E(y) = β₀ + β₁x and calculating the minimum Sum of Squared Error (SSE).

Explore how to create a linear regression model to analyze the relationship between the length of a conductor and the electric current. Learn the step-by-step calculation of slope (β₁), intercept (β₀), and the Sum of Squared Error (SSE) using the given dataset. Suitable for high school students studying advanced statistics.

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