document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178445>Question 1178445</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let Y have a chi-square distribution with 7 degrees of freedom. Find the following probabilities.<BR>\n" );
document.write( "(a) P (Y > y0) = 0.025<BR>\n" );
document.write( "(b) P (a < Y < b) = 0.90\r<BR>\n" );
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document.write( "(c) P (Y > 1.239).\r<BR>\n" );
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document.write( "thank you  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850347 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=850347\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve these probability problems using the chi-square distribution with 7 degrees of freedom.\r<BR>\n" );
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document.write( "**(a) P(Y > y0) = 0.025**\r<BR>\n" );
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document.write( "* We need to find the value y0 such that the area to the right of it under the chi-square curve is 0.025.<BR>\n" );
document.write( "* This is the 97.5th percentile of the chi-square distribution.<BR>\n" );
document.write( "* Using a chi-square table or calculator, we find y0.\r<BR>\n" );
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document.write( "**(b) P(a < Y < b) = 0.90**\r<BR>\n" );
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document.write( "* We need to find the values a and b such that the area between them under the chi-square curve is 0.90.<BR>\n" );
document.write( "* Since we want the middle 90%, we need to exclude the lower 5% and the upper 5%.<BR>\n" );
document.write( "* Thus, 'a' is the 5th percentile, and 'b' is the 95th percentile of the chi-square distribution.\r<BR>\n" );
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document.write( "**(c) P(Y > 1.239)**\r<BR>\n" );
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document.write( "* We need to find the area to the right of 1.239 under the chi-square curve.<BR>\n" );
document.write( "* This is equivalent to 1 - P(Y ≤ 1.239), where P(Y ≤ 1.239) is the cumulative distribution function (CDF) at 1.239.\r<BR>\n" );
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document.write( "**Calculations**\r<BR>\n" );
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document.write( "Using a chi-square calculator or statistical software:\r<BR>\n" );
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document.write( "**(a) P(Y > y0) = 0.025**\r<BR>\n" );
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document.write( "* y0 = 16.0128 (approximately)\r<BR>\n" );
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document.write( "**(b) P(a < Y < b) = 0.90**\r<BR>\n" );
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document.write( "* a = 2.1673 (approximately)<BR>\n" );
document.write( "* b = 14.0671 (approximately)\r<BR>\n" );
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document.write( "**(c) P(Y > 1.239)**\r<BR>\n" );
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document.write( "* P(Y ≤ 1.239) = 0.0100 (approximately)<BR>\n" );
document.write( "* P(Y > 1.239) = 1 - 0.0100 = 0.9900 (approximately)\r<BR>\n" );
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document.write( "**Results**\r<BR>\n" );
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document.write( "* **(a) y0 = 16.0128**<BR>\n" );
document.write( "* **(b) a = 2.1673, b = 14.0671**<BR>\n" );
document.write( "* **(c) P(Y > 1.239) = 0.9900**<BR>\n" );
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