document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179922>Question 1179922</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Doctors nationally believe that 70% of a certain type of operation are successful. In a particular hospital, 42 of these operations were observed and 32 of them were successful. At  is this hospital's success rate different from the national average? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850158 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=850158\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to conduct a hypothesis test to determine if the hospital's success rate is different from the national average:\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):** The hospital's success rate is equal to the national average (p = 0.70).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The hospital's success rate is different from the national average (p ≠ 0.70). This is a two-tailed test.\r<BR>\n" );
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document.write( "**2. Significance Level:** α = 0.05 (Assuming this common value if not provided)\r<BR>\n" );
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document.write( "**3. Calculate the Sample Proportion (p̂):**\r<BR>\n" );
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document.write( "* p̂ = (Number of successful operations) / (Total number of operations)<BR>\n" );
document.write( "* p̂ = 32 / 42 ≈ 0.7619\r<BR>\n" );
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document.write( "**4. Calculate the Test Statistic (z-score):**\r<BR>\n" );
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document.write( "z = (p̂ - p) / √(p(1 - p) / n)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* p̂ = sample proportion (0.7619)<BR>\n" );
document.write( "* p = hypothesized population proportion (0.70)<BR>\n" );
document.write( "* n = sample size (42)\r<BR>\n" );
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document.write( "z = (0.7619 - 0.70) / √(0.70 * (1 - 0.70) / 42)<BR>\n" );
document.write( "z = 0.0619 / √(0.70 * 0.30 / 42)<BR>\n" );
document.write( "z = 0.0619 / √(0.21 / 42)<BR>\n" );
document.write( "z = 0.0619 / √0.005<BR>\n" );
document.write( "z = 0.0619 / 0.0707<BR>\n" );
document.write( "z ≈ 0.875\r<BR>\n" );
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document.write( "**5. Determine the P-value:**\r<BR>\n" );
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document.write( "Since this is a two-tailed test, we need to find the probability of getting a z-score as extreme as 0.875 or -0.875. Using a z-table or calculator:\r<BR>\n" );
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document.write( "* P(z < -0.875) ≈ 0.1909<BR>\n" );
document.write( "* P(z > 0.875) ≈ 0.1909<BR>\n" );
document.write( "* P-value = 2 * 0.1909 ≈ 0.3818\r<BR>\n" );
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document.write( "**6. Make a Decision:**\r<BR>\n" );
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document.write( "Compare the p-value to the significance level (α):\r<BR>\n" );
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document.write( "* p-value (0.3818) > α (0.05)\r<BR>\n" );
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document.write( "Since the p-value is *greater than* the significance level, we *fail to reject* the null hypothesis.\r<BR>\n" );
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document.write( "**7. Conclusion:**\r<BR>\n" );
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document.write( "There is not sufficient evidence at the α = 0.05 level of significance to conclude that the hospital's success rate is different from the national average. Therefore, the hospital's success rate is not significantly different from the national average.<BR>\n" );
document.write( " </SPAN>\n" );
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