document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168831>Question 1168831</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a study of the accuracy of fast food​ drive-through orders, one restaurant had 31 orders that were not accurate among 356 orders observed. Use a  0.01significance level to test the claim that the rate of inaccurate orders is equal to​ 10%. Does the accuracy rate appear to be​ acceptable?<BR>\n" );
document.write( "Need to find the <BR>\n" );
document.write( "Hypothesis test <BR>\n" );
document.write( "p-value  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851510 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=851510\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You got it! Let's reiterate the solution for clarity and accuracy.\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 inaccurate orders is equal to 10% (p = 0.10).<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** The proportion of inaccurate orders is not equal to 10% (p ≠ 0.10).\r<BR>\n" );
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document.write( "This is a two-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) = 356<BR>\n" );
document.write( "* Number of inaccurate orders (x) = 31<BR>\n" );
document.write( "* Sample proportion (p̂) = x/n = 31 / 356 ≈ 0.08708\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 (0.08708)<BR>\n" );
document.write( "* p₀ = hypothesized proportion (0.10)<BR>\n" );
document.write( "* n = sample size (356)\r<BR>\n" );
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document.write( "$$ z = \frac{0.08708 - 0.10}{\sqrt{\frac{0.10(1-0.10)}{356}}} $$\r<BR>\n" );
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document.write( "$$ z = \frac{-0.01292}{\sqrt{\frac{0.09}{356}}} $$\r<BR>\n" );
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document.write( "$$ z = \frac{-0.01292}{\sqrt{0.000252808}} $$\r<BR>\n" );
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document.write( "$$ z = \frac{-0.01292}{0.0158999} $$\r<BR>\n" );
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document.write( "$$ z \approx -0.8126 $$\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 two-tailed test, we need to find the area in both tails beyond z = -0.8126 and z = 0.8126.\r<BR>\n" );
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document.write( "* P(Z < -0.8126) ≈ 0.2081<BR>\n" );
document.write( "* P(Z > 0.8126) ≈ 0.2081<BR>\n" );
document.write( "* P-value = 2 * 0.2081 ≈ 0.4162\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 (0.4162) with the significance level (0.01).<BR>\n" );
document.write( "* Since 0.4162 > 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 reject the claim that the rate of inaccurate orders is equal to 10%.\r<BR>\n" );
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document.write( "**Does the accuracy rate appear to be acceptable?**\r<BR>\n" );
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document.write( "The sample proportion of inaccurate orders (approximately 8.71%) is slightly lower than the claimed 10%. Statistically, we cannot reject the 10% rate.\r<BR>\n" );
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document.write( "* From the statistical test, there is no evidence that it is different than 10%.<BR>\n" );
document.write( "* Whether it is acceptable is up to the restaurant, and the customers. 8.7% is still a fairly high error rate, and could be deemed unaceptable, even though the test failed to reject the null hypothesis.\r<BR>\n" );
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document.write( "**Summary**\r<BR>\n" );
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document.write( "* **Hypothesis test:** Two-tailed z-test for proportions.<BR>\n" );
document.write( "* **p-value:** approximately 0.4162.<BR>\n" );
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