document.write( "Question 1177689: H0:P=0 HA:P>0 r=0.30 n=20 α=0.0\r
\n" ); document.write( "\n" ); document.write( "Test the above set of hypotheses
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Algebra.Com's Answer #850406 by CPhill(1959)\"\" \"About 
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You've given me the following information for a hypothesis test:\r
\n" ); document.write( "\n" ); document.write( "* **Null Hypothesis (H0):** P = 0
\n" ); document.write( "* **Alternative Hypothesis (HA):** P > 0
\n" ); document.write( "* **Sample Correlation Coefficient (r):** 0.30
\n" ); document.write( "* **Sample Size (n):** 20
\n" ); document.write( "* **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.)\r
\n" ); document.write( "\n" ); document.write( "**Understanding the Test**\r
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "**Test Statistic**\r
\n" ); document.write( "\n" ); document.write( "The test statistic for a correlation coefficient is calculated using the t-distribution:\r
\n" ); document.write( "\n" ); document.write( "* t = r * √(n - 2) / √(1 - r²)\r
\n" ); document.write( "\n" ); document.write( "**Calculations**\r
\n" ); document.write( "\n" ); document.write( "1. **Calculate the Test Statistic (t):**
\n" ); document.write( " * t = 0.30 * √(20 - 2) / √(1 - 0.30²)
\n" ); document.write( " * t = 0.30 * √18 / √(1 - 0.09)
\n" ); document.write( " * t = 0.30 * √18 / √0.91
\n" ); document.write( " * t = 0.30 * 4.2426 / 0.9539
\n" ); document.write( " * t = 1.27278 / 0.9539
\n" ); document.write( " * t ≈ 1.3343\r
\n" ); document.write( "\n" ); document.write( "2. **Degrees of Freedom (df):**
\n" ); document.write( " * df = n - 2 = 20 - 2 = 18\r
\n" ); document.write( "\n" ); document.write( "3. **Critical Value:**
\n" ); document.write( " * 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.
\n" ); document.write( " * 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.
\n" ); document.write( " * Since alpha is 0.0, no matter what t value we calculated, we would fail to reject the null hypothesis.\r
\n" ); document.write( "\n" ); document.write( "4. **P-value:**
\n" ); document.write( " * With α = 0.0, the p-value would have to be exactly zero for us to reject the null.
\n" ); document.write( " * Using a t-distribution calculator or table, the p-value for t ≈ 1.3343 and df = 18 is greater than 0.05.\r
\n" ); document.write( "\n" ); document.write( "**Conclusion**\r
\n" ); document.write( "\n" ); document.write( "* Due to alpha being 0.0, we fail to reject the null hypothesis.
\n" ); document.write( "* 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.
\n" ); document.write( "* **Therefore, there is not sufficient evidence to support the claim that the population correlation coefficient is greater than 0.**\r
\n" ); document.write( "\n" ); document.write( "**Important Note Regarding α = 0.0**\r
\n" ); document.write( "\n" ); document.write( "* 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.\r
\n" ); document.write( "\n" ); document.write( "If you can provide a correct alpha value, I can give a more useful conclusion.
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