document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178442>Question 1178442</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let Y have a chi-square distribution with 15 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) = 0.95<BR>\n" );
document.write( "(c) P (Y ≥ 22.307).\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 #850348 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=850348\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's find the probabilities for a chi-square distribution with 15 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 are looking for the value y0 such that 2.5% of the chi-square distribution is to the left of it.<BR>\n" );
document.write( "* This is the 2.5th percentile of the chi-square distribution with 15 degrees of freedom.\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 want to find the values a and b such that 90% of the chi-square distribution lies between them.<BR>\n" );
document.write( "* This means we want to exclude the lower 5% and the upper 5%.<BR>\n" );
document.write( "* Therefore, 'a' is the 5th percentile, and 'b' is the 95th percentile.\r<BR>\n" );
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document.write( "**(c) P(Y ≥ 22.307)**\r<BR>\n" );
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document.write( "* We want to find the probability that Y is greater than or equal to 22.307.<BR>\n" );
document.write( "* This is the area to the right of 22.307 under the chi-square curve.\r<BR>\n" );
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document.write( "**Calculations**\r<BR>\n" );
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document.write( "Using the `scipy.stats` library in Python:\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import scipy.stats as stats\r<BR>\n" );
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document.write( "df = 15  # Degrees of freedom\r<BR>\n" );
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document.write( "# (a)<BR>\n" );
document.write( "y0 = stats.chi2.ppf(0.025, df)<BR>\n" );
document.write( "print(f\"(a) y0 = {y0:.4f}\")\r<BR>\n" );
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document.write( "# (b)<BR>\n" );
document.write( "a = stats.chi2.ppf(0.05, df)<BR>\n" );
document.write( "b = stats.chi2.ppf(0.95, df)<BR>\n" );
document.write( "print(f\"(b) a = {a:.4f}, b = {b:.4f}\")\r<BR>\n" );
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document.write( "# (c)<BR>\n" );
document.write( "prob_c = 1 - stats.chi2.cdf(22.307, df)<BR>\n" );
document.write( "print(f\"(c) P(Y >= 22.307) = {prob_c:.4f}\")<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Results**\r<BR>\n" );
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document.write( "* **(a) y0 = 6.2621**<BR>\n" );
document.write( "* **(b) a = 7.2609, b = 24.9958**<BR>\n" );
document.write( "* **(c) P(Y ≥ 22.307) = 0.1000**<BR>\n" );
document.write( " </SPAN>\n" );
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