document.write( "Question 1168837: Suppose 242 subjects are treated with a drug that is used to treat pain and 51 of them developed nausea. Use a 0.05 significance level to test the claim that more than ​20% of users develop nausea.
\n" ); document.write( "Need to find hypotheis test
\n" ); document.write( "p-value
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Algebra.Com's Answer #851505 by CPhill(1959)\"\" \"About 
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Let's solve this hypothesis test step-by-step.\r
\n" ); document.write( "\n" ); document.write( "**1. Define the Hypotheses**\r
\n" ); document.write( "\n" ); document.write( "* **Null Hypothesis (H₀):** The proportion of users who develop nausea is 20% or less (p ≤ 0.20).
\n" ); document.write( "* **Alternative Hypothesis (H₁):** The proportion of users who develop nausea is more than 20% (p > 0.20).\r
\n" ); document.write( "\n" ); document.write( "This is a right-tailed test.\r
\n" ); document.write( "\n" ); document.write( "**2. Set the Significance Level**\r
\n" ); document.write( "\n" ); document.write( "* α = 0.05\r
\n" ); document.write( "\n" ); document.write( "**3. Calculate the Sample Proportion**\r
\n" ); document.write( "\n" ); document.write( "* Sample size (n) = 242
\n" ); document.write( "* Number of subjects with nausea (x) = 51
\n" ); document.write( "* Sample proportion (p̂) = x/n = 51/242 ≈ 0.2107\r
\n" ); document.write( "\n" ); document.write( "**4. Calculate the Test Statistic**\r
\n" ); document.write( "\n" ); document.write( "We'll use the z-test for proportions:\r
\n" ); document.write( "\n" ); document.write( "$$ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} $$\r
\n" ); document.write( "\n" ); document.write( "Where:\r
\n" ); document.write( "\n" ); document.write( "* p̂ = sample proportion (0.2107)
\n" ); document.write( "* p₀ = hypothesized proportion (0.20)
\n" ); document.write( "* n = sample size (242)\r
\n" ); document.write( "\n" ); document.write( "$$ z = \frac{0.2107 - 0.20}{\sqrt{\frac{0.20(1-0.20)}{242}}} $$\r
\n" ); document.write( "\n" ); document.write( "$$ z = \frac{0.0107}{\sqrt{\frac{0.16}{242}}} $$\r
\n" ); document.write( "\n" ); document.write( "$$ z = \frac{0.0107}{\sqrt{0.000661157}} $$\r
\n" ); document.write( "\n" ); document.write( "$$ z = \frac{0.0107}{0.02571297} $$\r
\n" ); document.write( "\n" ); document.write( "$$ z \approx 0.4161 $$\r
\n" ); document.write( "\n" ); document.write( "**5. Calculate the P-value**\r
\n" ); document.write( "\n" ); document.write( "Since this is a right-tailed test, we need to find the area to the right of z = 0.4161 in the standard normal distribution.\r
\n" ); document.write( "\n" ); document.write( "Using a z-table or calculator, we find:\r
\n" ); document.write( "\n" ); document.write( "* P(Z > 0.4161) = 1 - P(Z < 0.4161)
\n" ); document.write( "* P(Z < 0.4161) ≈ 0.6612
\n" ); document.write( "* P(Z > 0.4161) ≈ 1 - 0.6612 ≈ 0.3388\r
\n" ); document.write( "\n" ); document.write( "Therefore, the p-value is approximately 0.3388.\r
\n" ); document.write( "\n" ); document.write( "**6. Make a Decision**\r
\n" ); document.write( "\n" ); document.write( "* Compare the p-value (0.3388) with the significance level (0.05).
\n" ); document.write( "* Since 0.3388 > 0.05, we fail to reject the null hypothesis.\r
\n" ); document.write( "\n" ); document.write( "**7. Conclusion**\r
\n" ); document.write( "\n" ); document.write( "There is not sufficient evidence at the 0.05 significance level to support the claim that more than 20% of users develop nausea.\r
\n" ); document.write( "\n" ); document.write( "**Summary**\r
\n" ); document.write( "\n" ); document.write( "* **Hypothesis test:** Right-tailed z-test for proportions.
\n" ); document.write( "* **p-value:** approximately 0.3388.
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