document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192469>Question 1192469</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A  recent  article  in  a  leading  magazine  claimed  that  the  homemakers  spend  an average of 22 hours per week preparing meals. A director of marketing at a food-processing company surveyed that a recent sample of 28 homemakers resulted in an  average  time  preparation  of  meals  of  17.5  hours  per  week,  with  a  standard deviation of 4.3 hours. Can the director conclude that the average preparation of meals is lesser? Use 0.05 significance level. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848982 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=848982\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Set up the Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The average time homemakers spend preparing meals is 22 hours per week (μ = 22).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The average time homemakers spend preparing meals is less than 22 hours per week (μ < 22).\r<BR>\n" );
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document.write( "**2. Calculate the t-statistic**\r<BR>\n" );
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document.write( "* **Formula:** <BR>\n" );
document.write( "   t = (sample mean - population mean) / (sample standard deviation / √sample size)<BR>\n" );
document.write( "   t = (17.5 - 22) / (4.3 / √28) <BR>\n" );
document.write( "   t = -4.5 / 0.811 <BR>\n" );
document.write( "   t ≈ -5.55\r<BR>\n" );
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document.write( "**3. Determine the Degrees of Freedom**\r<BR>\n" );
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document.write( "* Degrees of Freedom (df) = sample size - 1 = 28 - 1 = 27\r<BR>\n" );
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document.write( "**4. Find the Critical Value**\r<BR>\n" );
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document.write( "* Since this is a one-tailed test (we're only interested in whether the mean is *less* than 22 hours), we'll use a one-tailed t-distribution table. <BR>\n" );
document.write( "* For a significance level of 0.05 and 27 degrees of freedom, the critical t-value is approximately -1.703.\r<BR>\n" );
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document.write( "**5. Compare t-statistic to Critical Value**\r<BR>\n" );
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document.write( "* The calculated t-statistic (-5.55) is less than the critical t-value (-1.703).\r<BR>\n" );
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document.write( "**6. Make a Decision**\r<BR>\n" );
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document.write( "* Since the t-statistic falls in the critical region, we **reject the null hypothesis**.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "At the 0.05 significance level, there is sufficient evidence to conclude that the average time homemakers spend preparing meals is less than 22 hours per week.<BR>\n" );
document.write( " </SPAN>\n" );
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