document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178776>Question 1178776</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A study claims that all homeowners in a town spend an average of 8 hours or<BR>\n" );
document.write( "more on house cleaning and gardening during a weekend. A researcher<BR>\n" );
document.write( "wanted to check if this claim is true. A random sample of 20 homeowners<BR>\n" );
document.write( "taken by this researcher showed that they spend an average of 7.68 hours on<BR>\n" );
document.write( "such chores during a weekend. The population of such times for all<BR>\n" );
document.write( "homeowners in this town is normally distributed with the population standard<BR>\n" );
document.write( "deviation of 2.1 hours. Using the 1% significance level, can you conclude that<BR>\n" );
document.write( "the claim that all homeowners spend an average of 8 hours or more on such<BR>\n" );
document.write( "chores during a weekend is false? [10 marks] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850297 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=850297\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's conduct a hypothesis test to determine if the claim is false.\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):** μ ≥ 8 (The average time spent is 8 hours or more)<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** μ < 8 (The average time spent is less than 8 hours)\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 (1%)\r<BR>\n" );
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document.write( "**3. Given Data:**\r<BR>\n" );
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document.write( "* Sample size (n): 20<BR>\n" );
document.write( "* Sample mean (x̄): 7.68 hours<BR>\n" );
document.write( "* Population standard deviation (σ): 2.1 hours\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 = (x̄ - μ) / (σ / √n)<BR>\n" );
document.write( "* z = (7.68 - 8) / (2.1 / √20)<BR>\n" );
document.write( "* z = -0.32 / (2.1 / 4.472)<BR>\n" );
document.write( "* z = -0.32 / 0.4696<BR>\n" );
document.write( "* z ≈ -0.6814\r<BR>\n" );
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document.write( "**5. Find the Critical Value:**\r<BR>\n" );
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document.write( "* For a left-tailed test with α = 0.01, the critical z-value is -2.33 (from a z-table or calculator).\r<BR>\n" );
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document.write( "**6. Make a Decision:**\r<BR>\n" );
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document.write( "* Compare the calculated z-score (-0.6814) to the critical z-value (-2.33).<BR>\n" );
document.write( "* Since -0.6814 > -2.33, the calculated z-score does not fall in the rejection region.<BR>\n" );
document.write( "* Therefore, we fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**7. Draw a Conclusion:**\r<BR>\n" );
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document.write( "* There is not sufficient evidence at the 1% significance level to conclude that the claim that all homeowners spend an average of 8 hours or more on such chores during a weekend is false.\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "The research does not provide enough evidence to reject the claim, so we cannot conclude that the claim is false.<BR>\n" );
document.write( " </SPAN>\n" );
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