document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167859>Question 1167859</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find P81, the 81-percentile. This is the temperature reading separating the bottom 81% from the top 19%.<BR>\n" );
document.write( "p81=? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852085 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=852085\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "from scipy.stats import norm\r<BR>\n" );
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document.write( "# Given parameters<BR>\n" );
document.write( "mean = 0  # degrees Celsius<BR>\n" );
document.write( "std_dev = 1.00  # degrees Celsius<BR>\n" );
document.write( "percentile = 81  # as a percentage\r<BR>\n" );
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document.write( "# Convert percentile to a decimal for the PPF function<BR>\n" );
document.write( "probability = percentile / 100\r<BR>\n" );
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document.write( "# Find the Z-score corresponding to the 81st percentile<BR>\n" );
document.write( "# norm.ppf(q, loc=0, scale=1) returns the inverse of the CDF, i.e., the value x<BR>\n" );
document.write( "# such that P(X <= x) = q, for a normal distribution with mean=loc and std_dev=scale.<BR>\n" );
document.write( "z_score_p81 = norm.ppf(probability)\r<BR>\n" );
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document.write( "# Convert the Z-score back to the temperature reading (x)<BR>\n" );
document.write( "# x = mean + Z * std_dev<BR>\n" );
document.write( "p81_temperature = mean + z_score_p81 * std_dev\r<BR>\n" );
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document.write( "print(f\"The Z-score for the 81st percentile is: {z_score_p81:.3f}\")<BR>\n" );
document.write( "print(f\"P81, the 81-percentile temperature reading is: {p81_temperature:.2f}°C\")<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "The Z-score for the 81st percentile is: 0.878<BR>\n" );
document.write( "P81, the 81-percentile temperature reading is: 0.88°C\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "To find $P\_{81}$, the 81st percentile temperature reading, we use the properties of the normal distribution.\r<BR>\n" );
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document.write( "Given:\r<BR>\n" );
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document.write( "  * Mean ($\\mu$) = $0^\\circ \\text{C}$<BR>\n" );
document.write( "  * Standard Deviation ($\\sigma$) = $1.00^\\circ \\text{C}$<BR>\n" );
document.write( "  * Percentile = 81st percentile, which means we are looking for the temperature reading ($x$) such that $81%$ of the readings are below it.\r<BR>\n" );
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document.write( "-- end list -->\r<BR>\n" );
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document.write( "1.  **Find the Z-score corresponding to the 81st percentile:**<BR>\n" );
document.write( "    The Z-score is the value on the standard normal distribution (mean = 0, standard deviation = 1) that corresponds to a cumulative probability of 0.81.<BR>\n" );
document.write( "    Using a Z-table or statistical software, the Z-score for $P(Z \\le z) = 0.81$ is approximately $0.878$.\r<BR>\n" );
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document.write( "2.  **Convert the Z-score back to the temperature reading:**<BR>\n" );
document.write( "    We use the formula: $x = \\mu + Z\\sigma$<BR>\n" );
document.write( "    $x = 0 + (0.878) \\times 1.00$<BR>\n" );
document.write( "    $x = 0.878$\r<BR>\n" );
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document.write( "Rounding to two decimal places, $P\_{81} = 0.88^\\circ \\text{C}$.\r<BR>\n" );
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document.write( "$P\_{81} = \\boxed{0.88}$ </SPAN>\n" );
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