document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179923>Question 1179923</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A political strategist claims that 56% of voters in Madison County support his candidate. In a poll of 300 randomly selected voters, 150 of them support the strategist's candidate. At  is the political strategist's claim warranted? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850131 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=850131\">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 strategist's claim is warranted:\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 proportion of voters who support the candidate is 56% (p = 0.56).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The proportion of voters who support the candidate is *not* 56% (p ≠ 0.56). This is a two-tailed test.\r<BR>\n" );
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document.write( "**2. Significance Level:** α = 0.05 (If not specified, we will assume this common value)\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 voters supporting the candidate) / (Total number of voters)<BR>\n" );
document.write( "* p̂ = 150 / 300 = 0.50\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 (0.50)<BR>\n" );
document.write( "* p = hypothesized population proportion (0.56)<BR>\n" );
document.write( "* n = sample size (300)\r<BR>\n" );
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document.write( "z = (0.50 - 0.56) / √(0.56 * (1 - 0.56) / 300)<BR>\n" );
document.write( "z = -0.06 / √(0.56 * 0.44 / 300)<BR>\n" );
document.write( "z = -0.06 / √(0.2464 / 300)<BR>\n" );
document.write( "z = -0.06 / √0.0008213<BR>\n" );
document.write( "z = -0.06 / 0.02866<BR>\n" );
document.write( "z ≈ -2.09\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 -2.09 or 2.09. Using a z-table or calculator:\r<BR>\n" );
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document.write( "* P(z < -2.09) ≈ 0.0183<BR>\n" );
document.write( "* P(z > 2.09) ≈ 0.0183<BR>\n" );
document.write( "* P-value = 2 * 0.0183 ≈ 0.0366\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 (0.0366) < α (0.05)\r<BR>\n" );
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document.write( "Since the p-value is *less than* the significance level, we *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 sufficient evidence at the α = 0.05 level of significance to conclude that the proportion of voters in Madison County who support the candidate is *not* 56%. Therefore, the political strategist's claim is *not* warranted.<BR>\n" );
document.write( " </SPAN>\n" );
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