Question 1198702
**a) Null Hypothesis (H0):** 

* μ₁ - μ₂ = 0 
    * Where: 
        * μ₁ is the population mean yield for Add1
        * μ₂ is the population mean yield for Add2

* This states that there is no difference in the mean yield of tomato plants between the two additives.

**b) Alternative Hypothesis (H1):**

* μ₁ - μ₂ ≠ 0 
    * This states that there is a significant difference in the mean yield of tomato plants between the two additives.

**c) Type of Test Statistic**

* Since the population standard deviations are unknown and unequal, we will use the **Welch's t-test**.

**d) Value of the Test Statistic**

* **Calculate the pooled variance (s_p²)** 
    * (Note: Since we are assuming unequal variances, we do not use the pooled variance)

* **Calculate the standard error:**
    * SE = √[(s₁²/n₁) + (s₂²/n₂)] 
    * SE = √[(112.1/12) + (1793.3/13)] 
    * SE ≈ 11.849

* **Calculate the t-statistic:**
    * t = (x̄₁ - x̄₂) / SE 
    * t = (133.7 - 158.2) / 11.849 
    * t ≈ -2.071

**e) Critical Values**

* **Degrees of Freedom (df):** 
    * Using the Welch-Satterthwaite equation for unequal variances:
        * df ≈ [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]² 
              / [(s₁²/n₁)²/(n₁-1)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)²/(n₂-1)] 
        * df ≈ [(112.1/12)²/(12-1) + (1793.3/13)²/(13-1)]² 
              / [(112.1/12)²/(12-1)²/(12-1) + (1793.3/13)²/(13-1)²/(13-1)] 
        * df ≈ 11.67 
    * We'll use df = 11 for the t-distribution table.

* **Critical Values (Two-tailed test at α = 0.10):** 
    * From the t-distribution table with 11 degrees of freedom and α/2 = 0.05, the critical values are approximately ±1.796.

**f) Conclusion**

* **Compare the test statistic to the critical values:**
    * Calculated t-statistic (-2.071) < Lower Critical Value (-1.796)

* **Decision:** Since the calculated t-statistic falls in the rejection region, we **reject the null hypothesis**.

* **Conclusion:** At the 0.10 level of significance, there is sufficient evidence to conclude that there is a statistically significant difference in the mean yield of tomato plants grown with Add1 and Add2.

**Note:**

* This analysis assumes that the populations of yields for both additives are normally distributed. 
* The large difference in sample variances might indicate that the assumption of equal variances may not be valid, which is why the Welch's t-test is used.
* Further investigation might be warranted to determine which additive leads to higher yields.