document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194224>Question 1194224</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 8.34 Working Women In an Advertising Age<BR>\n" );
document.write( "white paper concerning the changing role of women as<BR>\n" );
document.write( "“breadwinners” in the American family, it was<BR>\n" );
document.write( "reported that according to their survey with JWT,<BR>\n" );
document.write( "working men reported doing 54 minutes of household<BR>\n" );
document.write( "chores a day, while working women reported tackling<BR>\n" );
document.write( "72 minutes daily. But when examined more closely,<BR>\n" );
document.write( "Millennial men reported doing just as many household<BR>\n" );
document.write( "chores as the average working women, 72 minutes,<BR>\n" );
document.write( "compared to an average of 54 minutes among both<BR>\n" );
document.write( "Boomer men and Xer men.6 The information that follows is adapted from these data and is based on random samples of 1136 men and 795 women.<BR>\n" );
document.write( "Standard<BR>\n" );
document.write( "Mean Deviation n<BR>\n" );
document.write( "All Women 72 10.4 795<BR>\n" );
document.write( "All Men 54 12.7 1136<BR>\n" );
document.write( "Millennial 72 9.2 345<BR>\n" );
document.write( "Boomers 54 13.9 475<BR>\n" );
document.write( "Xers 54 10.5 316<BR>\n" );
document.write( "a. Construct a 95% confidence interval for the average<BR>\n" );
document.write( "time all men spend doing household chores.<BR>\n" );
document.write( "b. Construct a 95% confidence interval for the average<BR>\n" );
document.write( "time women spend doing household chores </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848464 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848464\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Construct a 95% Confidence Interval for the Average Time All Men Spend Doing Household Chores**\r<BR>\n" );
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document.write( "* **Given:**<BR>\n" );
document.write( "    * Sample Mean (x̄) = 54 minutes<BR>\n" );
document.write( "    * Sample Standard Deviation (s) = 12.7 minutes<BR>\n" );
document.write( "    * Sample Size (n) = 1136 men<BR>\n" );
document.write( "    * Confidence Level = 95% \r<BR>\n" );
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document.write( "* **Find the Critical Value (zα/2):**<BR>\n" );
document.write( "    * For a 95% confidence level, α = 0.05 <BR>\n" );
document.write( "    * zα/2 = 1.96 (from the standard normal distribution table)\r<BR>\n" );
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document.write( "* **Calculate the Standard Error of the Mean (SEM):**<BR>\n" );
document.write( "    * SEM = s / √n <BR>\n" );
document.write( "    * SEM = 12.7 / √1136 <BR>\n" );
document.write( "    * SEM ≈ 0.377\r<BR>\n" );
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document.write( "* **Calculate the Margin of Error (ME):**<BR>\n" );
document.write( "    * ME = zα/2 * SEM <BR>\n" );
document.write( "    * ME = 1.96 * 0.377 <BR>\n" );
document.write( "    * ME ≈ 0.739\r<BR>\n" );
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document.write( "* **Construct the Confidence Interval:**<BR>\n" );
document.write( "    * Lower Limit = x̄ - ME = 54 - 0.739 = 53.261 minutes<BR>\n" );
document.write( "    * Upper Limit = x̄ + ME = 54 + 0.739 = 54.739 minutes\r<BR>\n" );
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document.write( "* **95% Confidence Interval for All Men:** (53.261 minutes, 54.739 minutes)\r<BR>\n" );
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document.write( "**b. Construct a 95% Confidence Interval for the Average Time Women Spend Doing Household Chores**\r<BR>\n" );
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document.write( "* **Given:**<BR>\n" );
document.write( "    * Sample Mean (x̄) = 72 minutes<BR>\n" );
document.write( "    * Sample Standard Deviation (s) = 10.4 minutes<BR>\n" );
document.write( "    * Sample Size (n) = 795 women<BR>\n" );
document.write( "    * Confidence Level = 95% \r<BR>\n" );
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document.write( "* **Find the Critical Value (zα/2):**<BR>\n" );
document.write( "    * Same as part (a): zα/2 = 1.96\r<BR>\n" );
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document.write( "* **Calculate the Standard Error of the Mean (SEM):**<BR>\n" );
document.write( "    * SEM = s / √n <BR>\n" );
document.write( "    * SEM = 10.4 / √795 <BR>\n" );
document.write( "    * SEM ≈ 0.368\r<BR>\n" );
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document.write( "* **Calculate the Margin of Error (ME):**<BR>\n" );
document.write( "    * ME = zα/2 * SEM <BR>\n" );
document.write( "    * ME = 1.96 * 0.368 <BR>\n" );
document.write( "    * ME ≈ 0.721\r<BR>\n" );
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document.write( "* **Construct the Confidence Interval:**<BR>\n" );
document.write( "    * Lower Limit = x̄ - ME = 72 - 0.721 = 71.279 minutes<BR>\n" );
document.write( "    * Upper Limit = x̄ + ME = 72 + 0.721 = 72.721 minutes\r<BR>\n" );
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document.write( "* **95% Confidence Interval for All Women:** (71.279 minutes, 72.721 minutes)\r<BR>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "* We are 95% confident that the true average time all men spend on household chores lies between 53.261 and 54.739 minutes.<BR>\n" );
document.write( "* We are 95% confident that the true average time all women spend on household chores lies between 71.279 and 72.721 minutes.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );