document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191683>Question 1191683</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Using the data below, suppose we focus on the proportions of patients who show improvement. Is there a statistically significant difference in the proportions of patients who show improvement between treatments 1 and 2. Run the test at a 5% level of significance.\r<BR>\n" );
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document.write( "Symptoms\r<BR>\n" );
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document.write( "Worsened\r<BR>\n" );
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document.write( "No Effect\r<BR>\n" );
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document.write( "Symptoms Improved\r<BR>\n" );
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document.write( "Total\r<BR>\n" );
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document.write( "Treatment 1\r<BR>\n" );
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document.write( "22\r<BR>\n" );
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document.write( "14\r<BR>\n" );
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document.write( "14\r<BR>\n" );
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document.write( "50\r<BR>\n" );
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document.write( "Treatment 2\r<BR>\n" );
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document.write( "14\r<BR>\n" );
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document.write( "15\r<BR>\n" );
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document.write( "21\r<BR>\n" );
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document.write( "50\r<BR>\n" );
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document.write( "Treatment 3\r<BR>\n" );
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document.write( "9\r<BR>\n" );
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document.write( "12\r<BR>\n" );
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document.write( "29\r<BR>\n" );
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document.write( "50\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849146 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=849146\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to test for a statistically significant difference in improvement proportions between Treatment 1 and Treatment 2:\r<BR>\n" );
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document.write( "**1. State the Hypotheses:**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no difference in the proportions of patients who show improvement between Treatment 1 and Treatment 2.<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** There is a difference in the proportions of patients who show improvement between Treatment 1 and Treatment 2.\r<BR>\n" );
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document.write( "**2. Calculate Proportions:**\r<BR>\n" );
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document.write( "* **Treatment 1:** Proportion of improvement = 14/50 = 0.28<BR>\n" );
document.write( "* **Treatment 2:** Proportion of improvement = 21/50 = 0.42\r<BR>\n" );
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document.write( "**3. Perform a Two-Proportion Z-Test:**\r<BR>\n" );
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document.write( "This test is appropriate for comparing two proportions from independent samples.\r<BR>\n" );
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document.write( "* **Calculate the pooled proportion:**<BR>\n" );
document.write( "   p = (14 + 21) / (50 + 50) = 35/100 = 0.35\r<BR>\n" );
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document.write( "* **Calculate the standard error:**<BR>\n" );
document.write( "   SE = sqrt{ p * (1-p) * [(1/50) + (1/50)] }<BR>\n" );
document.write( "   SE = sqrt{ 0.35 * 0.65 * 0.04 } ≈ 0.095\r<BR>\n" );
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document.write( "* **Calculate the Z-statistic:**<BR>\n" );
document.write( "   Z = (0.42 - 0.28) / 0.095 ≈ 1.47\r<BR>\n" );
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document.write( "**4. Determine the p-value:**\r<BR>\n" );
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document.write( "Using a Z-table or calculator, find the p-value corresponding to a Z-statistic of 1.47 (two-tailed test). The p-value is approximately 0.14.\r<BR>\n" );
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document.write( "**5. Make a Decision:**\r<BR>\n" );
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document.write( "* **Significance Level (α):** 5% (0.05)\r<BR>\n" );
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document.write( "Since the p-value (0.14) is greater than the significance level (0.05), we fail to 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 not enough evidence to conclude that there is a statistically significant difference in the proportions of patients who show improvement between Treatment 1 and Treatment 2 at a 5% level of significance. </SPAN>\n" );
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