document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200426>Question 1200426</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A study was conducted to evaluate the hypothesis that tea consumption and premenstrual syndrome are associated. A total of 188 nursing students and 64 tea factory workers were given questionnaires. The prevalence of premenstrual syndrome was 39% among the nursing students<BR>\n" );
document.write( "and 77% among the tea factory workers. Calculate the 95% confidence interval for the prevalence of premenstrual syndrome for each of the two populations, nursing students and tea factory workers.<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 #848276 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=848276\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Nursing Students**\r<BR>\n" );
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document.write( "* **Sample Size (n1):** 188<BR>\n" );
document.write( "* **Prevalence of PMS (p1):** 39% = 0.39<BR>\n" );
document.write( "* **Number of Students with PMS (x1):** 188 * 0.39 = 73.32 ≈ 73 \r<BR>\n" );
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document.write( "* **Standard Error (SE1):** <BR>\n" );
document.write( "    * SE1 = √[p1 * (1 - p1) / n1] <BR>\n" );
document.write( "    * SE1 = √[0.39 * (1 - 0.39) / 188] <BR>\n" );
document.write( "    * SE1 ≈ 0.0394\r<BR>\n" );
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document.write( "* **95% Confidence Interval:** <BR>\n" );
document.write( "    * For a 95% confidence level, the z-score is 1.96.<BR>\n" );
document.write( "    * Lower Limit: p1 - (z * SE1) = 0.39 - (1.96 * 0.0394) ≈ 0.313 <BR>\n" );
document.write( "    * Upper Limit: p1 + (z * SE1) = 0.39 + (1.96 * 0.0394) ≈ 0.467\r<BR>\n" );
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document.write( "* **95% Confidence Interval for Nursing Students:** (0.313, 0.467)\r<BR>\n" );
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document.write( "**2. Tea Factory Workers**\r<BR>\n" );
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document.write( "* **Sample Size (n2):** 64<BR>\n" );
document.write( "* **Prevalence of PMS (p2):** 77% = 0.77<BR>\n" );
document.write( "* **Number of Students with PMS (x2):** 64 * 0.77 = 49.28 ≈ 49 \r<BR>\n" );
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document.write( "* **Standard Error (SE2):** <BR>\n" );
document.write( "    * SE2 = √[p2 * (1 - p2) / n2] <BR>\n" );
document.write( "    * SE2 = √[0.77 * (1 - 0.77) / 64] <BR>\n" );
document.write( "    * SE2 ≈ 0.0534\r<BR>\n" );
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document.write( "* **95% Confidence Interval:** <BR>\n" );
document.write( "    * Lower Limit: p2 - (z * SE2) = 0.77 - (1.96 * 0.0534) ≈ 0.665 <BR>\n" );
document.write( "    * Upper Limit: p2 + (z * SE2) = 0.77 + (1.96 * 0.0534) ≈ 0.875\r<BR>\n" );
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document.write( "* **95% Confidence Interval for Tea Factory Workers:** (0.665, 0.875)\r<BR>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "* We are 95% confident that the true proportion of nursing students experiencing PMS lies between 31.3% and 46.7%.<BR>\n" );
document.write( "* We are 95% confident that the true proportion of tea factory workers experiencing PMS lies between 66.5% and 87.5%.\r<BR>\n" );
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document.write( "These confidence intervals provide a range within which the true population proportions are likely to fall. \r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* These calculations assume that the samples are representative of the respective populations.<BR>\n" );
document.write( "* Larger sample sizes would generally lead to narrower confidence intervals and more precise estimates.<BR>\n" );
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