Question 1201174: The germination rate of seeds is defined as the proportion of seeds that when properly planted and watered, spruot and grow. A certain variety of grass seed usually has a germination rate of 0.80, and a company wants to see if spraying the seeds with a chemical that is known to change germination rates in other species will change the germination rate of this grass species.
(a) Suppose the company plans to spray a random sample of 400 seeds and conduct a two-sided test of H0: p=0.8 using (Alpha)=O.05. They determine that the power of this test against the alternative p=0.75 is 0.69. Interpret the power of this test.
(b) Describe two ways the company can increase the power of the test. What a disadvantage of each of these ways?
(c) The company researchers spray 400 seeds with the chemical and 307 of the seeds germinate. This produces a 95% confidence interval for the proportions of seeds that germinate of (0.726,0.809). Use this confidence interval to determine whether this test would reject or fail to reject the null hypothesis. Explain your reasoning.
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **a) Interpretation of Power**
* **Power:** The power of a statistical test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true.
* **In this context:** The power of 0.69 means that if the true germination rate after spraying the chemical is actually 0.75 (the alternative hypothesis), there is a 69% chance that the test will correctly detect this difference and reject the null hypothesis (that the germination rate is 0.80).
**b) Ways to Increase Power**
1. **Increase Sample Size:**
* **How:** Increase the number of seeds sprayed with the chemical.
* **Disadvantage:** Increased cost and time associated with testing more seeds.
2. **Increase Significance Level (α):**
* **How:** Increase α from 0.05 to a higher value (e.g., 0.10).
* **Disadvantage:** Increases the probability of Type I error (rejecting the null hypothesis when it is actually true).
**c) Decision Based on Confidence Interval**
* **The 95% confidence interval for the true germination rate is (0.726, 0.809).**
* **The null hypothesis states that the true germination rate is 0.80.**
* **Since the value 0.80 (the hypothesized value) falls within the 95% confidence interval,** we **fail to reject the null hypothesis.**
* **Reasoning:** The confidence interval suggests that the true germination rate could plausibly be 0.80. If the true rate were significantly different from 0.80, we would expect the confidence interval to not include 0.80.
**In summary:**
* The power of the test indicates the likelihood of correctly detecting a true difference in germination rates if it exists.
* Increasing sample size or significance level can increase power but also has potential drawbacks.
* The 95% confidence interval provides evidence that the true germination rate might not be significantly different from the hypothesized 0.80, leading to the failure to reject the null hypothesis.
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