document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208909>Question 1208909</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A genetic experiment involving peas yielded one sample of offspring consisting of 441 green peas and 168 yellow peas. Use a 0.01 significance level to test the claim that under the same​ circumstances, 26​% of offspring peas will be yellow. Identify the null​ hypothesis, alternative​ hypothesis, test​ statistic, P-value, conclusion about the null​ hypothesis, and final conclusion that addresses the original claim. Use the​ P-value method and the normal distribution as an approximation to the binomial distribution. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849184 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=849184\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to conduct a hypothesis test for the proportion of yellow peas:\r<BR>\n" );
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document.write( "**1. Hypotheses:**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** p = 0.26  (The proportion of yellow peas is 26%.)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** p ≠ 0.26 (The proportion of yellow peas is *not* 26%.)  This is a two-tailed test.\r<BR>\n" );
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document.write( "**2. Sample Proportion:**\r<BR>\n" );
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document.write( "* Total peas = 441 green + 168 yellow = 609<BR>\n" );
document.write( "* Sample proportion of yellow peas (p̂) = 168/609 ≈ 0.2759\r<BR>\n" );
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document.write( "**3. Test Statistic (z-score):**\r<BR>\n" );
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document.write( "We'll use the normal distribution as an approximation to the binomial. The formula for the test statistic is:\r<BR>\n" );
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document.write( "z = (p̂ - p) / sqrt[(p(1-p))/n]\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* p̂ is the sample proportion<BR>\n" );
document.write( "* p is the hypothesized proportion under H0<BR>\n" );
document.write( "* n is the sample size\r<BR>\n" );
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document.write( "z = (0.2759 - 0.26) / sqrt[(0.26 * 0.74)/609]<BR>\n" );
document.write( "z = 0.0159 / sqrt(0.000317)<BR>\n" );
document.write( "z ≈ 2.82\r<BR>\n" );
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document.write( "**4. 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 of the standard normal distribution that is beyond our z-score of 2.82.  Using a z-table or calculator:\r<BR>\n" );
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document.write( "P(z > 2.82) ≈ 0.0024<BR>\n" );
document.write( "P(z < -2.82) ≈ 0.0024\r<BR>\n" );
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document.write( "P-value = 2 * 0.0024 = 0.0048\r<BR>\n" );
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document.write( "**5. Conclusion about the Null Hypothesis:**\r<BR>\n" );
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document.write( "* Significance level (α) = 0.01\r<BR>\n" );
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document.write( "Since the P-value (0.0048) is *less than* the significance level (0.01), we *reject* the null hypothesis.\r<BR>\n" );
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document.write( "**6. Final Conclusion:**\r<BR>\n" );
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document.write( "There is sufficient evidence at the 0.01 significance level to reject the claim that 26% of offspring peas will be yellow.  The sample data suggests that the true proportion of yellow peas is different from 26%.<BR>\n" );
document.write( " </SPAN>\n" );
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