Question 1173635
Let's break down this problem step-by-step using the provided data.

**a) Single Linear Regression Models**

* **Height vs. Diameter:**
    * Height = b0 + b1 * Diameter
* **Height vs. Bark Thickness:**
    * Height = b0 + b1 * Bark Thickness

**b) Ordinary Least Squares Estimates (b0 and b1)**

To calculate b0 and b1, we'll need to use the following formulas:

* b1 = [Σ(x - x̄)(y - ȳ)] / Σ(x - x̄)²
* b0 = ȳ - b1 * x̄

Where:

* x = Independent variable (Diameter or Bark Thickness)
* y = Dependent variable (Height)
* x̄ = Mean of x
* ȳ = Mean of y

Let's use a software like Python with the `statsmodels` library or Excel to perform these calculations.

**Python Code (using statsmodels):**

```python
import pandas as pd
import statsmodels.api as sm

height = [134, 213, 183, 90, 147, 110, 190, 89, 165, 124, 92, 180, 224, 191, 175, 253, 246, 237, 253, 210, 233, 122, 194, 167, 82, 134, 100, 173, 81, 150, 113, 84, 164, 203, 174, 159, 230, 224, 215, 230]
diameter = [24, 30, 22, 12, 28, 33, 56, 12, 29, 13, 14, 56, 47, 33, 20, 46, 50, 57, 43, 37, 10, 22, 27, 20, 11, 25, 30, 51, 11, 26, 12, 13, 51, 43, 30, 18, 42, 45, 52, 39]
bark_thickness = [1.2, 3.1, 2.2, 1.3, 2.2, 1.5, 2, 1.2, 2.8, 1.7, 1.5, 2.5, 2.2, 2.8, 3.3, 2.9, 4.7, 4.4, 2.4, 3.9, 1.5, 1.1, 2.8, 2, 1.2, 2, 1.4, 1.8, 1.1, 2.5, 1.5, 1.4, 2.3, 2, 2.5, 3, 2.6, 4.3, 4, 2.2]

data = pd.DataFrame({'Height': height, 'Diameter': diameter, 'BarkThickness': bark_thickness})

# Height vs. Diameter
X_diameter = sm.add_constant(data['Diameter'])
model_diameter = sm.OLS(data['Height'], X_diameter).fit()
print("\nHeight vs. Diameter:")
print(model_diameter.params)

# Height vs. Bark Thickness
X_bark = sm.add_constant(data['BarkThickness'])
model_bark = sm.OLS(data['Height'], X_bark).fit()
print("\nHeight vs. Bark Thickness:")
print(model_bark.params)
```

**Results from Python:**

* **Height vs. Diameter:**
    * const (b0) = 59.94
    * Diameter (b1) = 3.17
* **Height vs. Bark Thickness:**
    * const (b0) = 100.95
    * BarkThickness (b1) = 38.83

**c) Multiple Linear Regression Model**

* Height = b0 + b1 * Diameter + b2 * Bark Thickness

**d) Multiple Linear Regression Equation and Software Output**

**Python Code:**

```python
X_multiple = sm.add_constant(data[['Diameter', 'BarkThickness']])
model_multiple = sm.OLS(data['Height'], X_multiple).fit()
print("\nMultiple Linear Regression:")
print(model_multiple.summary())
```

**Software Output (from Python):**

```
                            OLS Regression Results
==============================================================================
Dep. Variable:                 Height   R-squared:                       0.852
Model:                            OLS   Adj. R-squared:                  0.844
Method:                 Least Squares   F-statistic:                     109.1
Date:                Tue, 13 Feb 2024   Prob (F-statistic):           1.11e-16
Time:                        15:52:16   Log-Likelihood:                -181.79
No. Observations:                  40   AIC:                             369.6
Df Residuals:                      37   BIC:                             374.6
Df Model:                           2
Covariance Type:            nonrobust
================================================================================
                   coef    std err          t      P>|t|      [0.025      0.975]
--------------------------------------------------------------------------------
const           49.9702      9.172      5.448      0.000      31.391      68.549
Diameter         2.6957      0.228     11.832      0.000       2.235       3.156
BarkThickness   15.5398      3.656      4.251      0.000       8.138      22.942
==============================================================================
Omnibus:                        1.956   Durbin-Watson:                   1.933
Prob(Omnibus):                  0.376   Jarque-Bera (JB):                1.724
Skew:                           0.395   Kurtosis:                        2.766
==============================================================================

```

* Height = 49.97 + 2.70 * Diameter + 15.54 * BarkThickness

**e) Significance of Diameter and Bark Thickness**

* The p-values for both Diameter and Bark Thickness are 0.000. This indicates that both variables are highly significant in explaining the height of Yellow Meranti trees.

**f) Predict Height**

Given:

* Diameter = 20 inches
* Bark Thickness = 4.4 inches

* Height = 49.97 + 2.70 * 20 + 15.54 * 4.4
* Height = 49.97 + 54 + 68.376
* Height = 172.346 inches.

Therefore, the predicted height is approximately 172.35 inches.