document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174396>Question 1174396</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An electrical company claims that the average life of the bulbs it manufactures is 1200 hours within a standard deviation of 250 hours. If a random sample of 100 bulbs is chosen, what is the probability that the sample mean will be greater than 1150 hours </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850672 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=850672\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Calculate the Standard Error:**\r<BR>\n" );
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document.write( "* The standard error of the mean (SEM) measures the variability of the sample mean.<BR>\n" );
document.write( "* SEM = σ / √n<BR>\n" );
document.write( "    * σ = population standard deviation (250 hours)<BR>\n" );
document.write( "    * n = sample size (100 bulbs)<BR>\n" );
document.write( "* SEM = 250 / √100 = 250 / 10 = 25 hours\r<BR>\n" );
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document.write( "**2. Calculate the z-score:**\r<BR>\n" );
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document.write( "* The z-score tells us how many standard errors the sample mean is away from the population mean.<BR>\n" );
document.write( "* z = (x̄ - μ) / SEM<BR>\n" );
document.write( "    * x̄ = sample mean (1150 hours)<BR>\n" );
document.write( "    * μ = population mean (1200 hours)<BR>\n" );
document.write( "    * SEM = standard error of the mean (25 hours)<BR>\n" );
document.write( "* z = (1150 - 1200) / 25 = -50 / 25 = -2\r<BR>\n" );
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document.write( "**3. Find the Probability:**\r<BR>\n" );
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document.write( "* We want the probability that the sample mean is greater than 1150 hours. This is the same as finding the area to the right of z = -2 in the standard normal distribution.<BR>\n" );
document.write( "* Using a z-table or calculator, look up the probability for z = -2. You'll find a value of approximately 0.0228. <BR>\n" );
document.write( "* Since we want the area to the *right* of z = -2, subtract this value from 1: 1 - 0.0228 = 0.9772\r<BR>\n" );
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document.write( "**Therefore, the probability that the sample mean will be greater than 1150 hours is approximately 0.9772.** </SPAN>\n" );
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