document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191903>Question 1191903</A>: <I><SPAN STYLE=\"word-break: break-all;\"> for a normal distribution if the value of mean is 70 and standard deviation is 15.75 find the points <BR>\n" );
document.write( "(a) in which it contains 98% area between them<BR>\n" );
document.write( "(b) in which it contains 95% area between them<BR>\n" );
document.write( "(c) in which it contains 85% area between them<BR>\n" );
document.write( "(d) in which 60% area between them </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849143 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=849143\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the points for the given normal distribution:\r<BR>\n" );
document.write( "\n" );
document.write( "**Understanding Confidence Intervals**\r<BR>\n" );
document.write( "\n" );
document.write( "A confidence interval represents a range of values within which a certain percentage of data is expected to fall.  In a normal distribution, these intervals are symmetric around the mean.  We use z-scores to determine the boundaries of these intervals.\r<BR>\n" );
document.write( "\n" );
document.write( "**Calculations**\r<BR>\n" );
document.write( "\n" );
document.write( "The formula for calculating the confidence interval is:\r<BR>\n" );
document.write( "\n" );
document.write( "* Lower Bound = Mean - (Z-score * Standard Deviation)<BR>\n" );
document.write( "* Upper Bound = Mean + (Z-score * Standard Deviation)\r<BR>\n" );
document.write( "\n" );
document.write( "Where the Z-score corresponds to the desired confidence level.  Here's a breakdown for each percentage:\r<BR>\n" );
document.write( "\n" );
document.write( "**(a) 98% Area**\r<BR>\n" );
document.write( "\n" );
document.write( "* Z-score for 98% (two-tailed): Approximately 2.33 (You can find this using a Z-table or a statistical calculator).<BR>\n" );
document.write( "* Lower Bound: 70 - (2.33 * 15.75) ≈ 33.36<BR>\n" );
document.write( "* Upper Bound: 70 + (2.33 * 15.75) ≈ 106.64\r<BR>\n" );
document.write( "\n" );
document.write( "**(b) 95% Area**\r<BR>\n" );
document.write( "\n" );
document.write( "* Z-score for 95% (two-tailed): Approximately 1.96<BR>\n" );
document.write( "* Lower Bound: 70 - (1.96 * 15.75) ≈ 39.13<BR>\n" );
document.write( "* Upper Bound: 70 + (1.96 * 15.75) ≈ 100.87\r<BR>\n" );
document.write( "\n" );
document.write( "**(c) 85% Area**\r<BR>\n" );
document.write( "\n" );
document.write( "* Z-score for 85% (two-tailed): Approximately 1.44<BR>\n" );
document.write( "* Lower Bound: 70 - (1.44 * 15.75) ≈ 47.33<BR>\n" );
document.write( "* Upper Bound: 70 + (1.44 * 15.75) ≈ 92.67\r<BR>\n" );
document.write( "\n" );
document.write( "**(d) 60% Area**\r<BR>\n" );
document.write( "\n" );
document.write( "* Z-score for 60% (two-tailed): Approximately 0.84<BR>\n" );
document.write( "* Lower Bound: 70 - (0.84 * 15.75) ≈ 56.74<BR>\n" );
document.write( "* Upper Bound: 70 + (0.84 * 15.75) ≈ 83.26\r<BR>\n" );
document.write( "\n" );
document.write( "**Summary of Results**\r<BR>\n" );
document.write( "\n" );
document.write( "* **98%:**  The interval containing 98% of the area is approximately (33.36, 106.64).<BR>\n" );
document.write( "* **95%:**  The interval containing 95% of the area is approximately (39.13, 100.87).<BR>\n" );
document.write( "* **85%:**  The interval containing 85% of the area is approximately (47.33, 92.67).<BR>\n" );
document.write( "* **60%:**  The interval containing 60% of the area is approximately (56.74, 83.26).<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );