SOLUTION: An engineer working for a large agribusiness has developed two types of soil additives he calls Add1 and Add2. The engineer wants to test whether there is any difference between th

Question 1198702: An engineer working for a large agribusiness has developed two types of soil additives he calls Add1 and Add2. The engineer wants to test whether there is any difference between the two additives in the mean yield of tomato plants grown using these additives.
The engineer studies a random sample of 12
tomato plants grown using Add1 and a random sample of 13
tomato plants grown using Add2. (These samples are chosen independently.) The plants grown with Add1 have a sample mean yield of 133.7 tomatoes with a sample variance of 112.1. The plants grown with Add2 have a sample mean yield of 158.2 tomatoes with a sample variance of 1793.3.
Assume that the two populations of yields are approximately normally distributed. Can the engineer conclude, at the 0.10 level of significance, that there is a difference between the population mean of the yields of tomato plants grown with Add1 and the population mean of the yields of tomato plants grown with Add2?
a) The null hypothesis:
b) The alternative hypothesis:
c) The type of test statistic (Z, t, chi square, or f):
d) The Value of the test statistic (round to at least 3 decimal places):
e) The 2 critical values (round to at least 3 decimal places):
F). At 0.1level of significance, can the engineer conclude that there is a difference between the mean yield of tomato plants grown with Add1 and the mean yield of tomato plants grown with Add2?
Answer by textot(100) (Show Source): You can put this solution on YOUR website!
**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.

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