3. Using Table $$ G $$, find the values for $$ \chi_{ ext {left }}^{2} $$ and $$ \chi_{ ext {right }}^{2} $$ a. $$ \alpha=0.05, n=12 $$ b. $$ \alpha=0.10, n=20 $$ c. $$ \alpha=0.05, n=27 $$ d. $$ \alpha=0.01, n=6 $$ e. $$ \alpha=0.10, n=41 $$

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3. Using Table $$ G $$, find the values for $$ \chi_{	ext {left }}^{2} $$ and $$ \chi_{	ext {right }}^{2} $$ a. $$ \alpha=0.05, n=12 $$ b. $$ \alpha=0.10, n=20 $$ c. $$ \alpha=0.05, n=27 $$ d. $$ \alpha=0.01, n=6 $$ e. $$ \alpha=0.10, n=41 $$

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To determine the critical values $$ \chi_{\text{left}}^{2} $$ and $$ \chi_{\text{right}}^{2} $$ for each scenario, we'll use the chi-square distribution table (referred to as Table $$ G $$). The degrees of freedom (df) are typically calculated as $$ \text{df} = n - 1 $$, where $$ n $$ is the sample size. Here's a step-by-step guide and the corresponding critical values for each part:

### General Steps:
1. **Determine Degrees of Freedom (df):**
   $$
   \text{df} = n - 1
   $$
2. **Find Critical Values:**
   - **Right Critical Value ($$ \chi_{\text{right}}^{2} $$)** corresponds to the upper tail (1 - $$ \alpha $$) of the chi-square distribution.
   - **Left Critical Value ($$ \chi_{\text{left}}^{2} $$)** corresponds to the lower tail ($$ \alpha $$) of the chi-square distribution.

### Detailed Solutions:

#### a. $$ \alpha = 0.05 $$, $$ n = 12 $$
- **Degrees of Freedom:**
  $$
  \text{df} = 12 - 1 = 11
  $$
- **Critical Values:**
  - $$ \chi_{\text{left}}^{2} $$ (0.05) ≈ **3.816**
  - $$ \chi_{\text{right}}^{2} $$ (0.95) ≈ **19.675**

#### b. $$ \alpha = 0.10 $$, $$ n = 20 $$
- **Degrees of Freedom:**
  $$
  \text{df} = 20 - 1 = 19
  $$
- **Critical Values:**
  - $$ \chi_{\text{left}}^{2} $$ (0.10) ≈ **10.117**
  - $$ \chi_{\text{right}}^{2} $$ (0.90) ≈ **28.845**

#### c. $$ \alpha = 0.05 $$, $$ n = 27 $$
- **Degrees of Freedom:**
  $$
  \text{df} = 27 - 1 = 26
  $$
- **Critical Values:**
  - $$ \chi_{\text{left}}^{2} $$ (0.05) ≈ **14.611**
  - $$ \chi_{\text{right}}^{2} $$ (0.95) ≈ **37.652**

#### d. $$ \alpha = 0.01 $$, $$ n = 6 $$
- **Degrees of Freedom:**
  $$
  \text{df} = 6 - 1 = 5
  $$
- **Critical Values:**
  - $$ \chi_{\text{left}}^{2} $$ (0.01) ≈ **1.145**
  - $$ \chi_{\text{right}}^{2} $$ (0.99) ≈ **15.086**

#### e. $$ \alpha = 0.10 $$, $$ n = 41 $$
- **Degrees of Freedom:**
  $$
  \text{df} = 41 - 1 = 40
  $$
- **Critical Values:**
  - $$ \chi_{\text{left}}^{2} $$ (0.10) ≈ **24.433**
  - $$ \chi_{\text{right}}^{2} $$ (0.90) ≈ **55.758**

### Summary of Critical Values:

| Part | $$ \alpha $$ | $$ n $$ | Degrees of Freedom (df) | $$ \chi_{\text{left}}^{2} $$ | $$ \chi_{\text{right}}^{2} $$ |
|------|--------------|---------|--------------------------|-------------------------------|--------------------------------|
| a    | 0.05         | 12      | 11                       | 3.816                         | 19.675                         |
| b    | 0.10         | 20      | 19                       | 10.117                        | 28.845                         |
| c    | 0.05         | 27      | 26                       | 14.611                        | 37.652                         |
| d    | 0.01         | 6       | 5                        | 1.145                         | 15.086                         |
| e    | 0.10         | 41      | 40                       | 24.433                        | 55.758                         |

**Note:** The critical values provided are approximate and based on standard chi-square distribution tables. For precise calculations, especially for large degrees of freedom or exact significance levels, it's recommended to use statistical software or more detailed tables.
Here are the critical values for each scenario: - **a.** $$ \alpha = 0.05 $$, $$ n = 12 $$: - $$ \chi_{\text{left}}^{2} \approx 3.816 $$ - $$ \chi_{\text{right}}^{2} \approx 19.675 $$ - **b.** $$ \alpha = 0.10 $$, $$ n = 20 $$: - $$ \chi_{\text{left}}^{2} \approx 10.117 $$ - $$ \chi_{\text{right}}^{2} \approx 28.845 $$ - **c.** $$ \alpha = 0.05 $$, $$ n = 27 $$: - $$ \chi_{\text{left}}^{2} \approx 14.611 $$ - $$ \chi_{\text{right}}^{2} \approx 37.652 $$ - **d.** $$ \alpha = 0.01 $$, $$ n = 6 $$: - $$ \chi_{\text{left}}^{2} \approx 1.145 $$ - $$ \chi_{\text{right}}^{2} \approx 15.086 $$ - **e.** $$ \alpha = 0.10 $$, $$ n = 41 $$: - $$ \chi_{\text{left}}^{2} \approx 24.433 $$ - $$ \chi_{\text{right}}^{2} \approx 55.758 $$

3. Using Table \( G \), find the values for \( \chi_{\text {left }}^{2} \) and \( \chi_{\text {right }}^{2} \) a. \( \alpha=0.05, n=12 \) b. \( \alpha=0.10, n=20 \) c. \( \alpha=0.05, n=27 \) d. \( \alpha=0.01, n=6 \) e. \( \alpha=0.10, n=41 \)

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