document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198525>Question 1198525</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The administrator of a local hospital has told the governing board that 30% of its <BR>\n" );
document.write( "emergency room patients are not really in need of emergency treatment. In checking <BR>\n" );
document.write( "a random sample of 400 emergency room patients, a board member finds that 35% <BR>\n" );
document.write( "of those treated were not true emergency cases. Using an appropriate hypothesis test <BR>\n" );
document.write( "and the 0.05 level, evaluate the administrator’s statement. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848292 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=848292\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** p = 0.30 (The true proportion of patients not needing emergency treatment is 30%)<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** p ≠ 0.30 (The true proportion of patients not needing emergency treatment is different from 30%) \r<BR>\n" );
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document.write( "**2. Calculate Sample Proportion**\r<BR>\n" );
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document.write( "* Sample Proportion (p̂): 35% = 0.35\r<BR>\n" );
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document.write( "**3. Calculate Test Statistic (z-score)**\r<BR>\n" );
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document.write( "* z = (p̂ - p₀) / √[p₀ * (1 - p₀) / n] <BR>\n" );
document.write( "    * where:<BR>\n" );
document.write( "        * p̂ = sample proportion (0.35)<BR>\n" );
document.write( "        * p₀ = hypothesized population proportion (0.30)<BR>\n" );
document.write( "        * n = sample size (400)\r<BR>\n" );
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document.write( "* z = (0.35 - 0.30) / √[0.30 * (1 - 0.30) / 400] <BR>\n" );
document.write( "* z = 0.05 / √[0.30 * 0.70 / 400] <BR>\n" );
document.write( "* z = 0.05 / √[0.21 / 400] <BR>\n" );
document.write( "* z = 0.05 / 0.0229 <BR>\n" );
document.write( "* z ≈ 2.18\r<BR>\n" );
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document.write( "**4. Determine Critical Values**\r<BR>\n" );
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document.write( "* Since this is a two-tailed test (H₁: p ≠ 0.30) at a 0.05 significance level, we need to find the critical values that divide the distribution into two tails with 0.025 area in each.<BR>\n" );
document.write( "* Using a standard normal distribution table, the critical values are approximately ±1.96.\r<BR>\n" );
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document.write( "**5. Make a Decision**\r<BR>\n" );
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document.write( "* **Compare the test statistic to the critical values:**<BR>\n" );
document.write( "    * Our calculated z-score (2.18) is greater than the critical value (1.96).\r<BR>\n" );
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document.write( "* **Decision:** Since the test statistic falls in the rejection region, we **reject the null hypothesis**.\r<BR>\n" );
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document.write( "**6. Conclusion**\r<BR>\n" );
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document.write( "* There is sufficient evidence at the 0.05 significance level to reject the administrator's claim that 30% of emergency room patients are not true emergencies. The sample data suggests that the actual proportion may be higher.\r<BR>\n" );
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document.write( "**In Summary:**\r<BR>\n" );
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document.write( "The board member's findings suggest that a higher proportion of patients may not be true emergencies than the administrator's claim. This information can be valuable for hospital resource allocation and improving emergency room efficiency.<BR>\n" );
document.write( " </SPAN>\n" );
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