document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202958>Question 1202958</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The weights of suitcases taken by travelers on international journeys are known to follow a distribution that is skewed to the right with mean = 18kg and standard deviation = 5kg. A certain flight has total weight limit on suitcases of 3500 kg. What is the probability that this weight limit will be exceeded if the total number of suitcases on board is :<BR>\n" );
document.write( "(a) 190?<BR>\n" );
document.write( "(b) 200?<BR>\n" );
document.write( "(c) 202?<BR>\n" );
document.write( "(d) 204?<BR>\n" );
document.write( "(e) 205? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847998 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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=847998\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define Variables**\r<BR>\n" );
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document.write( "* Let X_i represent the weight of the i-th suitcase.<BR>\n" );
document.write( "* Let S_n represent the sum of the weights of n suitcases: S_n = X_1 + X_2 + ... + X_n\r<BR>\n" );
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document.write( "**2. Determine the Mean and Standard Deviation of the Total Weight**\r<BR>\n" );
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document.write( "* **Mean of S_n:** <BR>\n" );
document.write( "    * E(S_n) = E(X_1) + E(X_2) + ... + E(X_n) = n * E(X_i) <BR>\n" );
document.write( "    * E(S_n) = n * 18 kg \r<BR>\n" );
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document.write( "* **Standard Deviation of S_n:** <BR>\n" );
document.write( "    * Var(S_n) = Var(X_1) + Var(X_2) + ... + Var(X_n) (assuming independence of suitcase weights)<BR>\n" );
document.write( "    * Var(S_n) = n * Var(X_i) = n * (5 kg)² = 25n <BR>\n" );
document.write( "    * Standard Deviation of S_n: σ_S_n = √(Var(S_n)) = √(25n) = 5√n\r<BR>\n" );
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document.write( "**3. Calculate the Probability of Exceeding the Weight Limit**\r<BR>\n" );
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document.write( "* We need to find P(S_n > 3500 kg) for each value of n.\r<BR>\n" );
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document.write( "* **Standardize the Total Weight:**<BR>\n" );
document.write( "    * Z = (S_n - E(S_n)) / σ_S_n <BR>\n" );
document.write( "    * Z = (S_n - 18n) / (5√n)\r<BR>\n" );
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document.write( "* **Find the Critical Value (z_c):**<BR>\n" );
document.write( "    * The weight limit is exceeded when S_n > 3500 kg.<BR>\n" );
document.write( "    * We need to find the z-score (z_c) corresponding to the probability of exceeding the weight limit.\r<BR>\n" );
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document.write( "* **Calculate the Probability:**<BR>\n" );
document.write( "    * P(S_n > 3500) = P(Z > z_c) \r<BR>\n" );
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document.write( "**4. Calculations for Different Numbers of Suitcases**\r<BR>\n" );
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document.write( "* **(a) n = 190:**<BR>\n" );
document.write( "    * E(S_190) = 190 * 18 = 3420 kg<BR>\n" );
document.write( "    * σ_S_190 = 5√190 ≈ 68.56 kg<BR>\n" );
document.write( "    * z_c = (3500 - 3420) / 68.56 ≈ 1.16<BR>\n" );
document.write( "    * P(S_190 > 3500) = P(Z > 1.16) <BR>\n" );
document.write( "        * Use a standard normal distribution table to find this probability.\r<BR>\n" );
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document.write( "* **(b) n = 200:**<BR>\n" );
document.write( "    * E(S_200) = 200 * 18 = 3600 kg<BR>\n" );
document.write( "    * σ_S_200 = 5√200 ≈ 70.71 kg<BR>\n" );
document.write( "    * z_c = (3500 - 3600) / 70.71 ≈ -1.41<BR>\n" );
document.write( "    * P(S_200 > 3500) = P(Z > -1.41) = 1 - P(Z ≤ -1.41)\r<BR>\n" );
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document.write( "* **(c) n = 202:**<BR>\n" );
document.write( "    * E(S_202) = 202 * 18 = 3636 kg<BR>\n" );
document.write( "    * σ_S_202 = 5√202 ≈ 71.13 kg<BR>\n" );
document.write( "    * z_c = (3500 - 3636) / 71.13 ≈ -1.91<BR>\n" );
document.write( "    * P(S_202 > 3500) = P(Z > -1.91) = 1 - P(Z ≤ -1.91)\r<BR>\n" );
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document.write( "* **(d) n = 204:**<BR>\n" );
document.write( "    * E(S_204) = 204 * 18 = 3672 kg<BR>\n" );
document.write( "    * σ_S_204 = 5√204 ≈ 71.41 kg<BR>\n" );
document.write( "    * z_c = (3500 - 3672) / 71.41 ≈ -2.41<BR>\n" );
document.write( "    * P(S_204 > 3500) = P(Z > -2.41) = 1 - P(Z ≤ -2.41)\r<BR>\n" );
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document.write( "* **(e) n = 205:**<BR>\n" );
document.write( "    * E(S_205) = 205 * 18 = 3690 kg<BR>\n" );
document.write( "    * σ_S_205 = 5√205 ≈ 71.49 kg<BR>\n" );
document.write( "    * z_c = (3500 - 3690) / 71.49 ≈ -2.66<BR>\n" );
document.write( "    * P(S_205 > 3500) = P(Z > -2.66) = 1 - P(Z ≤ -2.66)\r<BR>\n" );
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document.write( "**Use a standard normal distribution table or statistical software to find the probabilities associated with the calculated z-scores for each case.**\r<BR>\n" );
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document.write( "**Important Notes:**\r<BR>\n" );
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document.write( "* This analysis assumes that the weights of the suitcases are independent and identically distributed.<BR>\n" );
document.write( "* The Central Limit Theorem suggests that the distribution of the total weight (S_n) will tend towards a normal distribution as the number of suitcases (n) increases, even though the individual suitcase weights are not normally distributed.<BR>\n" );
document.write( "* This calculation provides an approximation of the probability. <BR>\n" );
document.write( "* In reality, the weight distribution might not perfectly follow a normal distribution, which could affect the accuracy of the results.\r<BR>\n" );
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document.write( "I hope this comprehensive explanation helps! Let me know if you have any further questions.<BR>\n" );
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