document.write( "Question 1192440: consider two pain relieving drugs compared two independent samples of 1000 individuals each. suppose 750 of these individuals receiving drugs one and 800 of those receiving drug two construct 90% confidence interval for the percentage of 2 pain receiving pills?
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Algebra.Com's Answer #848984 by CPhill(1959)\"\" \"About 
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**1. Calculate the Sample Proportions**\r
\n" ); document.write( "\n" ); document.write( "* **Drug 1:**
\n" ); document.write( " * Sample proportion (p1) = 750 / 1000 = 0.75 \r
\n" ); document.write( "\n" ); document.write( "* **Drug 2:**
\n" ); document.write( " * Sample proportion (p2) = 800 / 1000 = 0.80\r
\n" ); document.write( "\n" ); document.write( "**2. Calculate the Standard Error**\r
\n" ); document.write( "\n" ); document.write( "* **Pooled Proportion (p̂):**
\n" ); document.write( " * p̂ = (Number of successes in both samples) / (Total sample size)
\n" ); document.write( " * p̂ = (750 + 800) / (1000 + 1000) = 1550 / 2000 = 0.775\r
\n" ); document.write( "\n" ); document.write( "* **Standard Error (SE):**
\n" ); document.write( " * SE = √[p̂ * (1 - p̂) * (1/n1 + 1/n2)]
\n" ); document.write( " * SE = √[0.775 * (1 - 0.775) * (1/1000 + 1/1000)]
\n" ); document.write( " * SE = √[0.775 * 0.225 * (2/1000)]
\n" ); document.write( " * SE ≈ 0.0186\r
\n" ); document.write( "\n" ); document.write( "**3. Determine the Z-score for 90% Confidence Level**\r
\n" ); document.write( "\n" ); document.write( "* For a 90% confidence interval, the Z-score is 1.645.\r
\n" ); document.write( "\n" ); document.write( "**4. Calculate the Margin of Error**\r
\n" ); document.write( "\n" ); document.write( "* Margin of Error (ME) = Z-score * SE
\n" ); document.write( "* ME = 1.645 * 0.0186
\n" ); document.write( "* ME ≈ 0.0306\r
\n" ); document.write( "\n" ); document.write( "**5. Construct the Confidence Intervals**\r
\n" ); document.write( "\n" ); document.write( "* **Drug 1:**
\n" ); document.write( " * Lower limit: p1 - ME = 0.75 - 0.0306 = 0.7194
\n" ); document.write( " * Upper limit: p1 + ME = 0.75 + 0.0306 = 0.7806
\n" ); document.write( " * 90% CI for Drug 1: (0.7194, 0.7806)\r
\n" ); document.write( "\n" ); document.write( "* **Drug 2:**
\n" ); document.write( " * Lower limit: p2 - ME = 0.80 - 0.0306 = 0.7694
\n" ); document.write( " * Upper limit: p2 + ME = 0.80 + 0.0306 = 0.8306
\n" ); document.write( " * 90% CI for Drug 2: (0.7694, 0.8306)\r
\n" ); document.write( "\n" ); document.write( "**Interpretation:**\r
\n" ); document.write( "\n" ); document.write( "* We are 90% confident that the true proportion of individuals who experience pain relief with Drug 1 lies between 71.94% and 78.06%.
\n" ); document.write( "* We are 90% confident that the true proportion of individuals who experience pain relief with Drug 2 lies between 76.94% and 83.06%.\r
\n" ); document.write( "\n" ); document.write( "**Note:**\r
\n" ); document.write( "\n" ); document.write( "* These confidence intervals provide a range of plausible values for the true population proportions.
\n" ); document.write( "* The intervals do not overlap, suggesting that there might be a statistically significant difference in the effectiveness of the two drugs. However, further statistical analysis (such as a hypothesis test) would be needed to confirm this.\r
\n" ); document.write( "\n" ); document.write( "**Disclaimer:** \r
\n" ); document.write( "\n" ); document.write( "* This analysis provides a basic framework for constructing confidence intervals.
\n" ); document.write( "* In real-world scenarios, more sophisticated statistical methods might be necessary, especially when dealing with medical data.
\n" ); document.write( "* This information should not be considered medical advice.
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