document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180571>Question 1180571</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The U.S. Bureau of Labor Statistics reports that the average annual expenditure on food<BR>\n" );
document.write( "and drink for all families is $5700 (Money, December 2003). Assume that annual expenditure<BR>\n" );
document.write( "on food and drink is normally distributed and that the standard deviation is $1500.<BR>\n" );
document.write( "a. What is the range of expenditures of the 10% of families with the lowest annual spending<BR>\n" );
document.write( "on food and drink?<BR>\n" );
document.write( "b. What percentage of families spend more than $7000 annually on food and drink?<BR>\n" );
document.write( "c. What is the range of expenditures for the 5% of families with the highest annual spending<BR>\n" );
document.write( "on food and drink? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850106 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=850106\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**a. Range of expenditures for the lowest 10%:**\r<BR>\n" );
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document.write( "1.  **Find the z-score:** We're looking for the 10th percentile of the distribution. Use a z-table or calculator to find the z-score that corresponds to 0.10 (or 10%) of the area to the *left* of the mean. This z-score is approximately -1.28.\r<BR>\n" );
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document.write( "2.  **Convert to expenditure:** Use the z-score formula to convert this to a dollar amount:\r<BR>\n" );
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document.write( "    x = μ + zσ<BR>\n" );
document.write( "    x = $5700 + (-1.28)($1500)<BR>\n" );
document.write( "    x = $5700 - $1920<BR>\n" );
document.write( "    x = $3780\r<BR>\n" );
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document.write( "So, the range of expenditures for the lowest 10% of families is from $0 to $3780.\r<BR>\n" );
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document.write( "**b. Percentage of families spending more than $7000:**\r<BR>\n" );
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document.write( "1.  **Calculate the z-score:**\r<BR>\n" );
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document.write( "    z = (x - μ) / σ<BR>\n" );
document.write( "    z = ($7000 - $5700) / $1500<BR>\n" );
document.write( "    z = $1300 / $1500<BR>\n" );
document.write( "    z ≈ 0.87\r<BR>\n" );
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document.write( "2.  **Find the probability:** Use a z-table or calculator to find the area to the *right* of z = 0.87. This represents the percentage of families spending more than $7000.  P(z > 0.87) is approximately 0.1922 or 19.22%.\r<BR>\n" );
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document.write( "**c. Range of expenditures for the highest 5%:**\r<BR>\n" );
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document.write( "1.  **Find the z-score:** We're looking for the 95th percentile (since we want the top 5%). Use a z-table or calculator to find the z-score that corresponds to 0.95 (or 95%) of the area to the *left* of the mean. This z-score is approximately 1.645.\r<BR>\n" );
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document.write( "2.  **Convert to expenditure:**\r<BR>\n" );
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document.write( "    x = μ + zσ<BR>\n" );
document.write( "    x = $5700 + (1.645)($1500)<BR>\n" );
document.write( "    x = $5700 + $2467.50<BR>\n" );
document.write( "    x = $8167.50\r<BR>\n" );
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document.write( "So, the range of expenditures for the highest 5% of families is from $8167.50 and above.\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "*   a. The range of expenditures for the lowest 10% of families is from $0 to $3780.<BR>\n" );
document.write( "*   b. Approximately 19.22% of families spend more than $7000 annually on food and drink.<BR>\n" );
document.write( "*   c. The range of expenditures for the highest 5% of families is from $8167.50 and above.<BR>\n" );
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