document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181346>Question 1181346</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a previous poll, 29% of adult Americans with children under the age of 18 reported that their family ate dinner together seven nights a week. Suppose that, in a more recent poll, 318 of 1157 adults with children under the age of 18 reported that their family ate dinner together seven nights a week. Is there sufficient evidence that the proportion of families with children under the age of 18 who eat dinner together seven nights a week has decreased? Use the α = 0.10 level of significance. \r<BR>\n" );
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document.write( "[Remember, you should include your H0 , H1 , your p-value, and a conclusion statement explaining whether you accept or reject the claim.]\r<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 #850060 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=850060\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to conduct a hypothesis test for this 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 families who eat dinner together seven nights a week is the same as the previous poll (p = 0.29).<BR>\n" );
document.write( "*   **Alternative Hypothesis (H1):** The proportion of families who eat dinner together seven nights a week has *decreased* (p < 0.29). This is a left-tailed test.\r<BR>\n" );
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document.write( "**2. Significance Level:** α = 0.10\r<BR>\n" );
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document.write( "**3. Calculate the Sample Proportion:**\r<BR>\n" );
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document.write( "*   Sample proportion (p̂) = (Number of successes) / (Sample size)<BR>\n" );
document.write( "*   p̂ = 318 / 1157<BR>\n" );
document.write( "*   p̂ ≈ 0.275\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( "z = (0.275 - 0.29) / √(0.29 * (1 - 0.29) / 1157)<BR>\n" );
document.write( "z = -0.015 / √(0.2059 / 1157)<BR>\n" );
document.write( "z ≈ -0.015 / 0.0134<BR>\n" );
document.write( "z ≈ -1.12\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 left-tailed test, the p-value is the probability of getting a z-score as extreme as -1.12 or *lower*. Using a z-table or calculator:\r<BR>\n" );
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document.write( "P(z < -1.12) ≈ 0.1314\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 (0.1314) > α (0.10)\r<BR>\n" );
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document.write( "Since the p-value is *greater* than the significance level, 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 α = 0.10 level of significance to conclude that the proportion of families with children under the age of 18 who eat dinner together seven nights a week has decreased.<BR>\n" );
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