document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168838>Question 1168838</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider a drug testing company that provides a test for marijuana usage. Among 311 tested​ subjects, results from 25 subjects were wrong​ (either a false positive or a false​ negative). Use a 0.01 significance level to test the claim that less than 10 percent of the test results are wrong.<BR>\n" );
document.write( "Need to find <BR>\n" );
document.write( "Hypothesis test <BR>\n" );
document.write( "p-value<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 #851504 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=851504\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely, let's break down this hypothesis test step-by-step.\r<BR>\n" );
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document.write( "**1. Define the Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** The proportion of wrong test results is 10% or more (p ≥ 0.10).<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** The proportion of wrong test results is less than 10% (p < 0.10).\r<BR>\n" );
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document.write( "This is a left-tailed test.\r<BR>\n" );
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document.write( "**2. Set the Significance Level**\r<BR>\n" );
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document.write( "* α = 0.01\r<BR>\n" );
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document.write( "**3. Calculate the Sample Proportion**\r<BR>\n" );
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document.write( "* Sample size (n) = 311<BR>\n" );
document.write( "* Number of wrong results (x) = 25<BR>\n" );
document.write( "* Sample proportion (p̂) = x/n = 25/311 ≈ 0.0804\r<BR>\n" );
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document.write( "**4. Calculate the Test Statistic**\r<BR>\n" );
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document.write( "We'll use the z-test for proportions:\r<BR>\n" );
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document.write( "$$ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} $$\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* p̂ = sample proportion (0.0804)<BR>\n" );
document.write( "* p₀ = hypothesized proportion (0.10)<BR>\n" );
document.write( "* n = sample size (311)\r<BR>\n" );
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document.write( "$$ z = \frac{0.0804 - 0.10}{\sqrt{\frac{0.10(1-0.10)}{311}}} $$\r<BR>\n" );
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document.write( "$$ z = \frac{-0.0196}{\sqrt{\frac{0.09}{311}}} $$\r<BR>\n" );
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document.write( "$$ z = \frac{-0.0196}{\sqrt{0.000289389}} $$\r<BR>\n" );
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document.write( "$$ z = \frac{-0.0196}{0.01701144} $$\r<BR>\n" );
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document.write( "$$ z \approx -1.1522 $$\r<BR>\n" );
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document.write( "**5. Calculate the P-value**\r<BR>\n" );
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document.write( "Since this is a left-tailed test, we need to find the area to the left of z = -1.1522 in the standard normal distribution.\r<BR>\n" );
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document.write( "Using a z-table or calculator, we find:\r<BR>\n" );
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document.write( "* P(Z < -1.1522) ≈ 0.1246\r<BR>\n" );
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document.write( "Therefore, the p-value is approximately 0.1246.\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 (0.1246) with the significance level (0.01).<BR>\n" );
document.write( "* Since 0.1246 > 0.01, 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 0.01 significance level to support the claim that less than 10 percent of the test results are wrong.\r<BR>\n" );
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document.write( "**Summary**\r<BR>\n" );
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document.write( "* **Hypothesis test:** Left-tailed z-test for proportions.<BR>\n" );
document.write( "* **p-value:** approximately 0.1246.<BR>\n" );
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