document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185957>Question 1185957</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A market research consultant hired by the Pepsi-Cola Co, is interested in knowing if the proportion of consumers who favor Pepsi-Cola over Coke Classic is different than 50%. A random sample of 250 consumers from the market under investigation were polled and 464 preferred pepsi in a taste test. Using a 10 Level of significance, test the appropriate hypothesis to help answer his question </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849502 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=849502\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to conduct a hypothesis test for a proportion:\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 consumers who prefer Pepsi is 50%. (p = 0.50)<BR>\n" );
document.write( "*   **Alternative Hypothesis (H₁):** The proportion of consumers who prefer Pepsi is *not* 50%. (p ≠ 0.50)  This is a two-tailed test.\r<BR>\n" );
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document.write( "**2. Determine the Level of Significance:**\r<BR>\n" );
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document.write( "α = 0.10 (10%)\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 who prefer Pepsi) / (Total sample size)<BR>\n" );
document.write( "p̂ = 146 / 250 = 0.584\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) / sqrt[p(1-p) / n]<BR>\n" );
document.write( "z = (0.584 - 0.50) / sqrt[(0.50 * 0.50) / 250]<BR>\n" );
document.write( "z = 0.084 / sqrt(0.001)<BR>\n" );
document.write( "z = 0.084 / 0.0316<BR>\n" );
document.write( "z ≈ 2.66\r<BR>\n" );
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document.write( "**5. Determine the Critical Value(s) or P-value:**\r<BR>\n" );
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document.write( "*   **Critical Value Approach:** Since this is a two-tailed test with α = 0.10, we need to find the z-scores that correspond to the outer 5% of the distribution (2.5% in each tail).  Using a z-table or calculator, the critical values are approximately ±1.645.\r<BR>\n" );
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document.write( "*   **P-value Approach:** The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated (in this case, a z-score of 2.66), assuming the null hypothesis is true. Since it is a two-tailed test, we need to double the area in one tail.<BR>\n" );
document.write( "    P-value = 2 * P(z > 2.66) ≈ 2 * 0.0039 = 0.0078\r<BR>\n" );
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document.write( "**6. Make a Decision:**\r<BR>\n" );
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document.write( "*   **Critical Value Approach:** Since the calculated z-score (2.66) falls *outside* the critical region (-1.645 to +1.645), we *reject* the null hypothesis.\r<BR>\n" );
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document.write( "*   **P-value Approach:** Since the p-value (0.0078) is *less than* the level of significance (0.10), 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 10% level of significance to conclude that the proportion of consumers who prefer Pepsi is different from 50%.<BR>\n" );
document.write( " </SPAN>\n" );
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