document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200129>Question 1200129</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The resting heart rate for an adult horse should average about 𝜇 = 43 beats per minute with a (95% of data) range from 14 to 72 beats per minute. Let x be a random variable that represents the resting heart rate for an adult horse. Assume that x has a distribution that is approximately normal.\r<BR>\n" );
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document.write( "(a) The empirical rule indicates that for a symmetrical and bell-shaped distribution, approximately 95% of the data lies within two standard deviations of the mean. Therefore, a 95% range of data values extending from 𝜇  -  2𝜎 to 𝜇 + 2𝜎 is often used for \"commonly occurring\" data values. Note that the interval from 𝜇  -  2𝜎 to 𝜇 + 2𝜎 is 4𝜎 in length. This leads to a \"rule of thumb\" for estimating the standard deviation from a 95% range of data values:\r<BR>\n" );
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document.write( "For a symmetric, bell-shaped distribution:<BR>\n" );
document.write( "standard deviation ≈ range/4≈(high value  -  low value)/4 \r<BR>\n" );
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document.write( "where it is estimated that about 95% of the commonly occurring data values fall into this range.\r<BR>\n" );
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document.write( "Use this \"rule of thumb\" to estimate the standard deviation of x distribution. (Round your answer to one decimal place.)<BR>\n" );
document.write( " beats\r<BR>\n" );
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document.write( "(b) What is the probability that the heart rate is less than 25 beats per minute? (Round your answer to four decimal places.)\r<BR>\n" );
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document.write( "(c) What is the probability that the heart rate is greater than 60 beats per minute? (Round your answer to four decimal places.)\r<BR>\n" );
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document.write( "(d) What is the probability that the heart rate is between 25 and 60 beats per minute? (Round your answer to four decimal places.)\r<BR>\n" );
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document.write( "(e) A horse whose resting heart rate is in the upper 13% of the probability distribution of heart rates may have a secondary infection or illness that needs to be treated. What is the heart rate corresponding to the upper 13% cutoff point of the probability distribution? (Round your answer to the nearest whole number.)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848089 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=848089\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Estimate the Standard Deviation**\r<BR>\n" );
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document.write( "* **Use the \"rule of thumb\":** <BR>\n" );
document.write( "   Standard Deviation ≈ (High Value - Low Value) / 4 <BR>\n" );
document.write( "   Standard Deviation ≈ (72 beats/min - 14 beats/min) / 4 <BR>\n" );
document.write( "   Standard Deviation ≈ 58 beats/min / 4 <BR>\n" );
document.write( "   Standard Deviation ≈ 14.5 beats/min\r<BR>\n" );
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document.write( "**b) Probability of Heart Rate Less Than 25 bpm**\r<BR>\n" );
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document.write( "* **Standardize the value:**<BR>\n" );
document.write( "   z = (X - μ) / σ = (25 - 43) / 14.5 ≈ -1.24\r<BR>\n" );
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document.write( "* **Find the probability using a z-table:**<BR>\n" );
document.write( "   P(X < 25) = P(Z < -1.24) ≈ 0.1075\r<BR>\n" );
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document.write( "**c) Probability of Heart Rate Greater Than 60 bpm**\r<BR>\n" );
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document.write( "* **Standardize the value:**<BR>\n" );
document.write( "   z = (X - μ) / σ = (60 - 43) / 14.5 ≈ 1.17\r<BR>\n" );
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document.write( "* **Find the probability using a z-table:**<BR>\n" );
document.write( "   P(X > 60) = P(Z > 1.17) ≈ 1 - P(Z < 1.17) ≈ 1 - 0.8790 ≈ 0.1210\r<BR>\n" );
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document.write( "**d) Probability of Heart Rate Between 25 and 60 bpm**\r<BR>\n" );
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document.write( "* **Find the z-scores:**<BR>\n" );
document.write( "    * z1 = (25 - 43) / 14.5 ≈ -1.24 <BR>\n" );
document.write( "    * z2 = (60 - 43) / 14.5 ≈ 1.17\r<BR>\n" );
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document.write( "* **Find the probability using a z-table:**<BR>\n" );
document.write( "   P(25 < X < 60) = P(-1.24 < Z < 1.17) = P(Z < 1.17) - P(Z < -1.24) ≈ 0.8790 - 0.1075 ≈ 0.7715\r<BR>\n" );
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document.write( "**e) Heart Rate Corresponding to Upper 13%**\r<BR>\n" );
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document.write( "* **Find the z-score for the upper 13%:**<BR>\n" );
document.write( "    * Look up the z-score corresponding to the area to the left of 1 - 0.13 = 0.87 in a z-table. <BR>\n" );
document.write( "    * z ≈ 1.13\r<BR>\n" );
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document.write( "* **Calculate the heart rate:**<BR>\n" );
document.write( "    * X = μ + (z * σ) = 43 + (1.13 * 14.5) ≈ 61.4 \r<BR>\n" );
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document.write( "* **Round to the nearest whole number:** <BR>\n" );
document.write( "    * Heart Rate ≈ 61 beats/min\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* (a) Estimated Standard Deviation: 14.5 beats/min<BR>\n" );
document.write( "* (b) Probability of Heart Rate < 25 bpm: 0.1075<BR>\n" );
document.write( "* (c) Probability of Heart Rate > 60 bpm: 0.1210<BR>\n" );
document.write( "* (d) Probability of Heart Rate between 25 and 60 bpm: 0.7715<BR>\n" );
document.write( "* (e) Heart Rate for Upper 13%: 61 beats/min<BR>\n" );
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