document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184610>Question 1184610</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The diameter of a brand of tennis balls is approximately normally​ distributed, with a mean of 2.67 inches and a standard deviation of 0.05 inch. A random sample of 10 tennis balls is selected. Complete parts​ (a) through​ (d) below.\r<BR>\n" );
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document.write( " The probability is 60​% that the sample mean will be between what two values symmetrically distributed around the population​ mean?\r<BR>\n" );
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document.write( "The lower bound is----  inches\r<BR>\n" );
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document.write( "The upper bound is-----  inches.<BR>\n" );
document.write( "​(Round to two decimal places as​ needed.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849838 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=849838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the bounds for the sample mean:\r<BR>\n" );
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document.write( "1. **Find the z-score corresponding to the desired probability:**\r<BR>\n" );
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document.write( "Since we want the probability that the sample mean falls *between* two values symmetrically distributed around the population mean, and that probability is 60%, we need to find the z-scores that cut off the *remaining* 40% (100% - 60% = 40%). Since the distribution is symmetric, each tail will have 20% (40% / 2 = 20%).\r<BR>\n" );
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document.write( "Look up the z-score corresponding to 0.20 (or 20%) in a standard normal (z) table.  Since we are looking for the middle 60%, we need the z-score that corresponds to 0.70 (or 70%) or 1 - 0.20 on the left side of the distribution. The z-score is approximately 0.52.  (You can also use a calculator or statistical software to find this z-score more precisely, but 0.52 is generally sufficient for this type of problem.)\r<BR>\n" );
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document.write( "2. **Calculate the standard error of the mean:**\r<BR>\n" );
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document.write( "The standard error of the mean (SEM) is the standard deviation of the sample means.  It's calculated as:\r<BR>\n" );
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document.write( "SEM = σ / √n\r<BR>\n" );
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document.write( "where σ is the population standard deviation (0.05 inch) and n is the sample size (10).\r<BR>\n" );
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document.write( "SEM = 0.05 / √10 ≈ 0.0158 inch\r<BR>\n" );
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document.write( "3. **Calculate the margin of error:**\r<BR>\n" );
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document.write( "The margin of error is how much the sample mean is likely to vary from the population mean. It's calculated as:\r<BR>\n" );
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document.write( "Margin of Error = z * SEM\r<BR>\n" );
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document.write( "Margin of Error = 0.52 * 0.0158 ≈ 0.0082 inch\r<BR>\n" );
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document.write( "4. **Calculate the lower and upper bounds:**\r<BR>\n" );
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document.write( "* **Lower Bound:** Population Mean - Margin of Error = 2.67 - 0.0082 ≈ 2.66 inches<BR>\n" );
document.write( "* **Upper Bound:** Population Mean + Margin of Error = 2.67 + 0.0082 ≈ 2.68 inches\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "The lower bound is approximately 2.66 inches.<BR>\n" );
document.write( "The upper bound is approximately 2.68 inches.<BR>\n" );
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