document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191348>Question 1191348</A>: <I><SPAN STYLE=\"word-break: break-all;\"> \"An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed, with mean equal to 802 hours and a standard deviation of 93 hours. Find the probability that a random samples of 23 bulbs will have an average life between 793 and 817 hours.\"<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849222 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=849222\">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( "1. **Find 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:<BR>\n" );
document.write( "* σ is the population standard deviation (93 hours)<BR>\n" );
document.write( "* n is the sample size (23 bulbs)\r<BR>\n" );
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document.write( "SEM = 93 / √23 ≈ 19.38\r<BR>\n" );
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document.write( "2. **Calculate the z-scores:**\r<BR>\n" );
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document.write( "We need to convert the given average life values (793 and 817 hours) into z-scores.  The z-score tells us how many standard errors a particular sample mean is away from the population mean.\r<BR>\n" );
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document.write( "z = (x - μ) / SEM\r<BR>\n" );
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document.write( "Where:<BR>\n" );
document.write( "* x is the sample mean<BR>\n" );
document.write( "* μ is the population mean (802 hours)\r<BR>\n" );
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document.write( "* For x = 793 hours:<BR>\n" );
document.write( "   z₁ = (793 - 802) / 19.38 ≈ -0.47\r<BR>\n" );
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document.write( "* For x = 817 hours:<BR>\n" );
document.write( "   z₂ = (817 - 802) / 19.38 ≈ 0.78\r<BR>\n" );
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document.write( "3. **Find the probabilities:**\r<BR>\n" );
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document.write( "Use a z-table or calculator to find the area under the normal curve between these two z-scores.  This represents the probability that the sample mean will fall between 793 and 817 hours.\r<BR>\n" );
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document.write( "* Find the probability associated with z₂ = 0.78: P(z < 0.78) ≈ 0.7823<BR>\n" );
document.write( "* Find the probability associated with z₁ = -0.47: P(z < -0.47) ≈ 0.3192\r<BR>\n" );
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document.write( "4. **Calculate the probability between the two values:**\r<BR>\n" );
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document.write( "Subtract the smaller probability from the larger probability:\r<BR>\n" );
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document.write( "P(-0.47 < z < 0.78) = P(z < 0.78) - P(z < -0.47)<BR>\n" );
document.write( "P(-0.47 < z < 0.78) = 0.7823 - 0.3192 ≈ 0.4631\r<BR>\n" );
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document.write( "Therefore, the probability that a random sample of 23 bulbs will have an average life between 793 and 817 hours is approximately 0.4631 or 46.31%.<BR>\n" );
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