document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176817>Question 1176817</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Waiting period (minutes)l numbers of customersX l numbers of customersY<BR>\n" );
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document.write( "0 to but less than 5     | 8                   |6<BR>\n" );
document.write( "5 to but less than 10    | 10                  |5 <BR>\n" );
document.write( "10 to but less than 15   | 24                  |16<BR>\n" );
document.write( "15 to but less than 20   | 19                  |22 <BR>\n" );
document.write( "20 to but less than 25   | 17                  |18 <BR>\n" );
document.write( "25 to but less than 30   | 12                  |23<BR>\n" );
document.write( "_________________________|_____________________|____________________________\r<BR>\n" );
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document.write( "a) Calculate standard deviation and comment on the dispersion of the waiting time between counter X and Y.\r<BR>\n" );
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document.write( "b)Without calculate coefficient of skewness, comment on the shape of the distribution of the waiting time at counter X and counter Y, respectively.\r<BR>\n" );
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document.write( "c)Suggest the best measure of central tendency and best measure of dispersion for above data sets. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850562 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=850562\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the waiting time data for counters X and Y.\r<BR>\n" );
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document.write( "**a) Calculate Standard Deviation and Comment on Dispersion**\r<BR>\n" );
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document.write( "**1. Calculate the Midpoints**\r<BR>\n" );
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document.write( "* 0 to 5: 2.5<BR>\n" );
document.write( "* 5 to 10: 7.5<BR>\n" );
document.write( "* 10 to 15: 12.5<BR>\n" );
document.write( "* 15 to 20: 17.5<BR>\n" );
document.write( "* 20 to 25: 22.5<BR>\n" );
document.write( "* 25 to 30: 27.5\r<BR>\n" );
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document.write( "**2. Calculate Mean and Standard Deviation for Counter X**\r<BR>\n" );
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document.write( "* **Mean (X̄):**<BR>\n" );
document.write( "    * Σ(midpoint * frequency) / Σfrequency<BR>\n" );
document.write( "    * [(2.5 * 8) + (7.5 * 10) + (12.5 * 24) + (17.5 * 19) + (22.5 * 17) + (27.5 * 12)] / (8 + 10 + 24 + 19 + 17 + 12)<BR>\n" );
document.write( "    * [20 + 75 + 300 + 332.5 + 382.5 + 330] / 90<BR>\n" );
document.write( "    * 1440 / 90 = 16\r<BR>\n" );
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document.write( "* **Standard Deviation (sX):**<BR>\n" );
document.write( "    * sX = √[Σ(midpoint - X̄)² * frequency / (Σfrequency - 1)]<BR>\n" );
document.write( "    * sX = √[((2.5-16)²*8) + ((7.5-16)²*10) + ((12.5-16)²*24) + ((17.5-16)²*19) + ((22.5-16)²*17) + ((27.5-16)²*12)] / 89<BR>\n" );
document.write( "    * sX = √[(182.25 * 8) + (72.25 * 10) + (12.25 * 24) + (2.25 * 19) + (42.25 * 17) + (132.25 * 12)] / 89<BR>\n" );
document.write( "    * sX = √[1458 + 722.5 + 294 + 42.75 + 718.25 + 1587] / 89<BR>\n" );
document.write( "    * sX = √4822.5 / 89<BR>\n" );
document.write( "    * sX = √54.18539<BR>\n" );
document.write( "    * sX ≈ 7.36\r<BR>\n" );
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document.write( "**3. Calculate Mean and Standard Deviation for Counter Y**\r<BR>\n" );
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document.write( "* **Mean (Ȳ):**<BR>\n" );
document.write( "    * Σ(midpoint * frequency) / Σfrequency<BR>\n" );
document.write( "    * [(2.5 * 6) + (7.5 * 5) + (12.5 * 16) + (17.5 * 22) + (22.5 * 18) + (27.5 * 23)] / (6 + 5 + 16 + 22 + 18 + 23)<BR>\n" );
document.write( "    * [15 + 37.5 + 200 + 385 + 405 + 632.5] / 90<BR>\n" );
document.write( "    * 1675 / 90 = 18.61\r<BR>\n" );
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document.write( "* **Standard Deviation (sY):**<BR>\n" );
document.write( "    * sY = √[Σ(midpoint - Ȳ)² * frequency / (Σfrequency - 1)]<BR>\n" );
document.write( "    * sY = √[((2.5-18.61)²*6) + ((7.5-18.61)²*5) + ((12.5-18.61)²*16) + ((17.5-18.61)²*22) + ((22.5-18.61)²*18) + ((27.5-18.61)²*23)] / 89<BR>\n" );
document.write( "    * sY = √[(260.6721 * 6) + (123.4321 * 5) + (37.3281 * 16) + (1.2321 * 22) + (15.1281 * 18) + (79.0921 * 23)] / 89<BR>\n" );
document.write( "    * sY = √[1564.0326 + 617.1605 + 597.2496 + 27.1062 + 272.3058 + 1819.1183] / 89<BR>\n" );
document.write( "    * sY = √4896.9740 / 89<BR>\n" );
document.write( "    * sY = √55.02218<BR>\n" );
document.write( "    * sY ≈ 7.42\r<BR>\n" );
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document.write( "**Comment on Dispersion:**\r<BR>\n" );
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document.write( "* Counter X: Mean ≈ 16 minutes, Standard Deviation ≈ 7.36 minutes<BR>\n" );
document.write( "* Counter Y: Mean ≈ 18.61 minutes, Standard Deviation ≈ 7.42 minutes\r<BR>\n" );
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document.write( "Both counters have similar standard deviations, indicating comparable dispersion of waiting times. Counter Y has a slightly higher mean and standard deviation, suggesting slightly more variability and a longer average wait.\r<BR>\n" );
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document.write( "**b) Comment on the Shape of the Distribution (Without Coefficient of Skewness)**\r<BR>\n" );
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document.write( "* **Counter X:**<BR>\n" );
document.write( "    * The highest frequency (12.5 midpoint) is in the middle of the distribution.<BR>\n" );
document.write( "    * The frequencies decrease somewhat symmetrically on both sides.<BR>\n" );
document.write( "    * This suggests a roughly symmetrical or slightly skewed distribution.\r<BR>\n" );
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document.write( "* **Counter Y:**<BR>\n" );
document.write( "    * The highest frequencies are in the 15-20, 20-25, and 25-30 intervals.<BR>\n" );
document.write( "    * This suggests a right-skewed distribution, as there are more customers with longer waiting times.\r<BR>\n" );
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document.write( "**c) Suggest Best Measures of Central Tendency and Dispersion**\r<BR>\n" );
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document.write( "* **Central Tendency:**<BR>\n" );
document.write( "    * **Mean:** Appropriate for both distributions, especially if they are approximately symmetrical. However, if there is a strong skew, the median might be more representative.\r<BR>\n" );
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document.write( "* **Dispersion:**<BR>\n" );
document.write( "    * **Standard Deviation:** Appropriate for both distributions, as it measures the average spread of data around the mean.<BR>\n" );
document.write( "    * **Interquartile Range (IQR):** If there is a strong skew, the IQR might be a better measure of dispersion, as it is less affected by extreme values.\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* For both counters, the mean and standard deviation are suitable.<BR>\n" );
document.write( "* If counter Y's right skew is significant, the median and IQR might be preferred.<BR>\n" );
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