document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207924>Question 1207924</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of n = 36 observations has a mean x = 28.7 and a standard deviation s = 3.8.<BR>\n" );
document.write( "GIVEN:- <BR>\n" );
document.write( "- point estimate of the population mean 𝜇 = 28.7<BR>\n" );
document.write( "- 95% margin of error for your estimate = 1.2413<BR>\n" );
document.write( "- 90% confidence interval for 𝜇 = 27.658 to 29.742<BR>\n" );
document.write( "- In repeated sampling, 90% of all intervals constructed in this manner will enclose the population mean.<BR>\n" );
document.write( "- (a) bound is calculated using z𝛼, while the lower confidence limit in part (b) is calculated using z𝛼/2.<BR>\n" );
document.write( "    \r<BR>\n" );
document.write( "\n" );
document.write( "(a) Find a 90% lower confidence bound for the population mean 𝜇. (Round your answer to two decimal places.)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(b) How many observations do you need to estimate 𝜇 to within 0.6, with probability equal to 0.95? (Round your answer up to the nearest whole number.)<BR>\n" );
document.write( " observations<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 #848078 by </B><A HREF=/tutors/aboutme.mpl?userid=GingerAle><B>GingerAle(43)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=GingerAle><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848078\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Find a 90% lower confidence bound for the population mean 𝜇.**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Understand the Concept:**<BR>\n" );
document.write( "    * A lower confidence bound provides a minimum value for the population mean with a certain level of confidence. <BR>\n" );
document.write( "    * For a 90% lower bound, we're essentially saying that we are 90% confident that the population mean is greater than or equal to this value.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate the Lower Bound:**<BR>\n" );
document.write( "    * Since we're given the 90% confidence interval (27.658 to 29.742), the lower bound of this interval directly represents the 90% lower confidence bound for the population mean.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Answer:**<BR>\n" );
document.write( "    * The 90% lower confidence bound for the population mean 𝜇 is **27.66**.\r<BR>\n" );
document.write( "\n" );
document.write( "**b) Determine the sample size needed to estimate 𝜇 to within 0.6 with a 95% probability.**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Understand the Goal:**<BR>\n" );
document.write( "    * We want to find the sample size (n) that will ensure the margin of error (E) is 0.6 with a 95% confidence level.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Formula:**<BR>\n" );
document.write( "    * The margin of error (E) for a confidence interval for the mean is given by:<BR>\n" );
document.write( "        * E = zα/2 * (σ / √n) <BR>\n" );
document.write( "        * where:<BR>\n" );
document.write( "            * zα/2 is the critical value from the standard normal distribution for the desired confidence level (95% in this case)<BR>\n" );
document.write( "            * σ is the population standard deviation (we'll use the sample standard deviation 's' as an estimate)<BR>\n" );
document.write( "            * n is the sample size\r<BR>\n" );
document.write( "\n" );
document.write( "* **Rearrange the formula to solve for n:**<BR>\n" );
document.write( "    * n = (zα/2 * σ / E)²\r<BR>\n" );
document.write( "\n" );
document.write( "* **Find the critical value (zα/2):**<BR>\n" );
document.write( "    * For a 95% confidence level, zα/2 = 1.96 (from the standard normal distribution table)\r<BR>\n" );
document.write( "\n" );
document.write( "* **Plug in the values:**<BR>\n" );
document.write( "    * n = (1.96 * 3.8 / 0.6)² <BR>\n" );
document.write( "    * n = (12.32)² <BR>\n" );
document.write( "    * n ≈ 151.75\r<BR>\n" );
document.write( "\n" );
document.write( "* **Round up to the nearest whole number:**<BR>\n" );
document.write( "    * n = 152\r<BR>\n" );
document.write( "\n" );
document.write( "* **Answer:**<BR>\n" );
document.write( "    * You need **152 observations** to estimate 𝜇 to within 0.6 with a probability of 0.95.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );