SOLUTION: H0:P=0 HA:P>0 r=0.30 n=20 α=0.0 Test the above set of hypotheses

Question 1177689: H0:P=0 HA:P>0 r=0.30 n=20 α=0.0
Test the above set of hypotheses

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
You've given me the following information for a hypothesis test:
* **Null Hypothesis (H0):** P = 0
* **Alternative Hypothesis (HA):** P > 0
* **Sample Correlation Coefficient (r):** 0.30
* **Sample Size (n):** 20
* **Significance Level (α):** 0.0 (This is unusual, as α is typically a small positive value like 0.05 or 0.01. I'll address this below.)
**Understanding the Test**
This appears to be a hypothesis test for the population correlation coefficient (ρ). The hypotheses suggest a right-tailed test, meaning we're looking for evidence that the correlation is positive.
**Test Statistic**
The test statistic for a correlation coefficient is calculated using the t-distribution:
* t = r * √(n - 2) / √(1 - r²)
**Calculations**
1. **Calculate the Test Statistic (t):**
* t = 0.30 * √(20 - 2) / √(1 - 0.30²)
* t = 0.30 * √18 / √(1 - 0.09)
* t = 0.30 * √18 / √0.91
* t = 0.30 * 4.2426 / 0.9539
* t = 1.27278 / 0.9539
* t ≈ 1.3343
2. **Degrees of Freedom (df):**
* df = n - 2 = 20 - 2 = 18
3. **Critical Value:**
* Here's where the α = 0.0 causes a problem. When α = 0.0, it means you're requiring absolute certainty to reject the null hypothesis. This is impossible in statistical testing.
* With a positive alpha level, we would use a t-table or calculator to find the critical t-value for df = 18 and α. For example, if α = 0.05, the critical value would be approximately 1.734.
* Since alpha is 0.0, no matter what t value we calculated, we would fail to reject the null hypothesis.
4. **P-value:**
* With α = 0.0, the p-value would have to be exactly zero for us to reject the null.
* Using a t-distribution calculator or table, the p-value for t ≈ 1.3343 and df = 18 is greater than 0.05.
**Conclusion**
* Due to alpha being 0.0, we fail to reject the null hypothesis.
* If we had a positive alpha value, like 0.05, we would also fail to reject the null hypothesis, since the test statistic (1.3343) is less than the critical value (1.734) and the p-value is greater than 0.05.
* **Therefore, there is not sufficient evidence to support the claim that the population correlation coefficient is greater than 0.**
**Important Note Regarding α = 0.0**
* In practical statistics, α = 0.0 is almost never used. It implies that you will only reject the null hypothesis if there is absolutely no possibility of error, which is rarely achievable with real-world data.
If you can provide a correct alpha value, I can give a more useful conclusion.

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