document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174491>Question 1174491</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer of light bulbs produces bulbs that last a mean of 1050 hours with a standard deviation of 100 hours. What is the probability that the mean lifetime of a random sample of 10 of these bulbs is greater than 975 hours?\r<BR>\n" );
document.write( "\n" );
document.write( "The average public high school has 535 students with a standard deviation of 98.<BR>\n" );
document.write( "a.	If a public school is selected, what is the probability that the number of students enrolled is less than 550?\r<BR>\n" );
document.write( "\n" );
document.write( "b.	If a random sample of 25 public high schools is selected, what is the probability that the number of the students enrolled is greater than 600?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850641 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=850641\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down each part of this problem.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Light Bulb Lifetime**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Population Parameters:**<BR>\n" );
document.write( "    * Mean (μ) = 1050 hours<BR>\n" );
document.write( "    * Standard Deviation (σ) = 100 hours<BR>\n" );
document.write( "* **Sample Size:** n = 10<BR>\n" );
document.write( "* **Target Mean:** x̄ = 975 hours\r<BR>\n" );
document.write( "\n" );
document.write( "* **Central Limit Theorem:** Since the population standard deviation is known, we can use the z-distribution.<BR>\n" );
document.write( "* **Standard Error (SE):** SE = σ / √n = 100 / √10 ≈ 31.62<BR>\n" );
document.write( "* **Z-score:** z = (x̄ - μ) / SE = (975 - 1050) / 31.62 = -75 / 31.62 ≈ -2.37<BR>\n" );
document.write( "* **Probability:** P(x̄ > 975) = P(z > -2.37).<BR>\n" );
document.write( "    * Using a standard normal table or calculator, P(z > -2.37) ≈ 0.9911\r<BR>\n" );
document.write( "\n" );
document.write( "* **Answer:** The probability that the mean lifetime of a random sample of 10 bulbs is greater than 975 hours is approximately 0.9911.\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Public High School Students**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Population Parameters:**<BR>\n" );
document.write( "    * Mean (μ) = 535 students<BR>\n" );
document.write( "    * Standard Deviation (σ) = 98 students\r<BR>\n" );
document.write( "\n" );
document.write( "**(a) Probability for a Single School**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Target Students:** x = 550<BR>\n" );
document.write( "* **Z-score:** z = (x - μ) / σ = (550 - 535) / 98 = 15 / 98 ≈ 0.1531<BR>\n" );
document.write( "* **Probability:** P(x < 550) = P(z < 0.1531).<BR>\n" );
document.write( "    * Using a standard normal table or calculator, P(z < 0.1531) ≈ 0.5609\r<BR>\n" );
document.write( "\n" );
document.write( "* **Answer:** The probability that a public school has fewer than 550 students is approximately 0.5609.\r<BR>\n" );
document.write( "\n" );
document.write( "**(b) Probability for a Sample of 25 Schools**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Sample Size:** n = 25<BR>\n" );
document.write( "* **Target Mean:** x̄ = 600<BR>\n" );
document.write( "* **Standard Error (SE):** SE = σ / √n = 98 / √25 = 98 / 5 = 19.6<BR>\n" );
document.write( "* **Z-score:** z = (x̄ - μ) / SE = (600 - 535) / 19.6 = 65 / 19.6 ≈ 3.316<BR>\n" );
document.write( "* **Probability:** P(x̄ > 600) = P(z > 3.316).<BR>\n" );
document.write( "    * Using a standard normal table or calculator, P(z > 3.316) ≈ 0.00045\r<BR>\n" );
document.write( "\n" );
document.write( "* **Answer:** The probability that a random sample of 25 schools has a mean number of students greater than 600 is approximately 0.00045.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );