document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171900>Question 1171900</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The side effects of a new drug are being tested against a placebo. A simple random sample of 565 patients yields the results below. At a significance level of a= 0.10, is there enough evidence to conclude that the treatment is independent of the side effect of nausea?\r<BR>\n" );
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document.write( "Result	   Drug	   Placebo	Total<BR>\n" );
document.write( "Nausea	   36	    13	         49<BR>\n" );
document.write( "No nausea 254	    262	         516<BR>\n" );
document.write( "Total	  290	    275	         565\r<BR>\n" );
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document.write( "	The null hypothesis H_∅ :<BR>\n" );
document.write( "	The alternative hypothesis H_a :<BR>\n" );
document.write( "	The level of significance a=<BR>\n" );
document.write( "	The degrees of freedom:<BR>\n" );
document.write( "	The test statistic:<BR>\n" );
document.write( "	The critical value:<BR>\n" );
document.write( "	Decision:<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850834 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850834\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down this hypothesis test step-by-step.\r<BR>\n" );
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document.write( "**1. Setting up the Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** The treatment is independent of the side effect of nausea. In other words, there is no association between the drug and nausea.<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** The treatment is not independent of the side effect of nausea. There is an association between the drug and nausea.\r<BR>\n" );
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document.write( "**2. Level of Significance**\r<BR>\n" );
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document.write( "* α = 0.10 (given)\r<BR>\n" );
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document.write( "**3. Degrees of Freedom**\r<BR>\n" );
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document.write( "* Degrees of freedom (df) = (rows - 1) * (columns - 1)<BR>\n" );
document.write( "* In this case, df = (2 - 1) * (2 - 1) = 1 * 1 = 1\r<BR>\n" );
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document.write( "**4. Calculating the Test Statistic (Chi-Square)**\r<BR>\n" );
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document.write( "We'll use the chi-square (χ²) test for independence.\r<BR>\n" );
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document.write( "* **Observed Values (O):** The values in the table.<BR>\n" );
document.write( "* **Expected Values (E):** We need to calculate the expected values for each cell.<BR>\n" );
document.write( "    * Expected value = (row total * column total) / grand total\r<BR>\n" );
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document.write( "Here's how we calculate the expected values:\r<BR>\n" );
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document.write( "* E(Nausea, Drug) = (49 * 290) / 565 ≈ 25.15<BR>\n" );
document.write( "* E(Nausea, Placebo) = (49 * 275) / 565 ≈ 23.85<BR>\n" );
document.write( "* E(No nausea, Drug) = (516 * 290) / 565 ≈ 264.85<BR>\n" );
document.write( "* E(No nausea, Placebo) = (516 * 275) / 565 ≈ 251.15\r<BR>\n" );
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document.write( "Now, we calculate the chi-square statistic:\r<BR>\n" );
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document.write( "χ² = Σ [(O - E)² / E]\r<BR>\n" );
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document.write( "χ² = [(36 - 25.15)² / 25.15] + [(13 - 23.85)² / 23.85] + [(254 - 264.85)² / 264.85] + [(262 - 251.15)² / 251.15]\r<BR>\n" );
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document.write( "χ² ≈ [(10.85)² / 25.15] + [(-10.85)² / 23.85] + [(-10.85)² / 264.85] + [(10.85)² / 251.15]\r<BR>\n" );
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document.write( "χ² ≈ (117.7225 / 25.15) + (117.7225 / 23.85) + (117.7225 / 264.85) + (117.7225 / 251.15)\r<BR>\n" );
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document.write( "χ² ≈ 4.680 + 4.936 + 0.444 + 0.469\r<BR>\n" );
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document.write( "χ² ≈ 10.529\r<BR>\n" );
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document.write( "**5. Finding the Critical Value**\r<BR>\n" );
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document.write( "* Using a chi-square distribution table with df = 1 and α = 0.10, the critical value is 2.706.\r<BR>\n" );
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document.write( "**6. Decision**\r<BR>\n" );
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document.write( "* Compare the test statistic (10.529) with the critical value (2.706).<BR>\n" );
document.write( "* Since 10.529 > 2.706, the test statistic falls in the rejection region.<BR>\n" );
document.write( "* Therefore, we reject the null hypothesis.\r<BR>\n" );
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document.write( "**Summary**\r<BR>\n" );
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document.write( "* **H₀:** The treatment is independent of nausea.<BR>\n" );
document.write( "* **H₁:** The treatment is not independent of nausea.<BR>\n" );
document.write( "* **α:** 0.10<BR>\n" );
document.write( "* **df:** 1<BR>\n" );
document.write( "* **Test statistic (χ²):** 10.529<BR>\n" );
document.write( "* **Critical value:** 2.706<BR>\n" );
document.write( "* **Decision:** Reject the null hypothesis.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "There is enough evidence at the α = 0.10 significance level to conclude that the treatment is not independent of the side effect of nausea.<BR>\n" );
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