document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192499>Question 1192499</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Weight loss per week by adopting a particularly restrictive diet is normally <BR>\n" );
document.write( "distributed with a mean of 500 gms and a standard deviation of 2 gms. Weight <BR>\n" );
document.write( "loss per week by adopting an exercise regime is normally distributed with <BR>\n" );
document.write( "a mean of 200 gms and a standard deviation of 1 gm. What is the probability that <BR>\n" );
document.write( "Asha will lose 1kg in 1 week if she adopts both diet and exercise regime? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848981 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=848981\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define Variables**\r<BR>\n" );
document.write( "\n" );
document.write( "* Let X be the weight loss due to diet. <BR>\n" );
document.write( "* Let Y be the weight loss due to exercise.<BR>\n" );
document.write( "* Let Z be the total weight loss (Z = X + Y)\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Determine the Distribution of Z**\r<BR>\n" );
document.write( "\n" );
document.write( "* Since X and Y are both normally distributed, their sum (Z) is also normally distributed.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Mean of Z (μz):** <BR>\n" );
document.write( "    * μz = μx + μy <BR>\n" );
document.write( "    * μz = 500 gms + 200 gms = 700 gms\r<BR>\n" );
document.write( "\n" );
document.write( "* **Standard Deviation of Z (σz):**<BR>\n" );
document.write( "    * Since X and Y are independent, <BR>\n" );
document.write( "        * σz = √(σx² + σy²) <BR>\n" );
document.write( "        * σz = √(2² + 1²) = √5 gms \r<BR>\n" );
document.write( "\n" );
document.write( "**3. Standardize Z**\r<BR>\n" );
document.write( "\n" );
document.write( "* To find the probability, we need to standardize Z:<BR>\n" );
document.write( "    * Z-score = (Z - μz) / σz\r<BR>\n" );
document.write( "\n" );
document.write( "* For a weight loss of 1 kg (1000 gms):<BR>\n" );
document.write( "    * Z-score = (1000 gms - 700 gms) / √5 gms <BR>\n" );
document.write( "    * Z-score = 300 / √5 ≈ 134.16\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Find the Probability**\r<BR>\n" );
document.write( "\n" );
document.write( "* We need to find P(Z ≥ 1000 gms) <BR>\n" );
document.write( "* Since the Z-score is extremely high (134.16), the probability of losing 1 kg or more with both diet and exercise is **extremely close to 1 (or 100%)**. \r<BR>\n" );
document.write( "\n" );
document.write( "**In essence:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Combining a strict diet with an exercise regime leads to a significantly high probability of losing at least 1 kg per week. This is due to the combined effects of both methods and their relatively high mean weight loss and low standard deviations.\r<BR>\n" );
document.write( "\n" );
document.write( "**Important Note:** \r<BR>\n" );
document.write( "\n" );
document.write( "* This calculation assumes that the weight loss from diet and exercise are completely independent. In reality, there might be some interaction between the two.<BR>\n" );
document.write( "* This is a simplified model, and individual results may vary significantly.<BR>\n" );
document.write( "* Sustainable and healthy weight loss typically involves a combination of diet, exercise, and other lifestyle factors. \r<BR>\n" );
document.write( "\n" );
document.write( "I hope this helps! Let me know if you have any other questions.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );