document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170626>Question 1170626</A>: <I><SPAN STYLE=\"word-break: break-all;\">   -  Time Series and Non-Parametric Tests   <BR>\n" );
document.write( "1. The number of fishing rods selling each day is given below. Perform analyses of the time series to determine which model should be used for forecasting. (10 points) <BR>\n" );
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document.write( "a. 3 day moving average analysis<BR>\n" );
document.write( " b. 4 day moving average analysis<BR>\n" );
document.write( " c. 3 day weighted moving average analysis with weights w1=0.2, w2=0.3 and w3=0.5 with w1 on the oldest data<BR>\n" );
document.write( " d. exponential smoothing analysis with a = 0.3.<BR>\n" );
document.write( " e. Which model provides a better fit of the data?<BR>\n" );
document.write( " f. Forecast day 13 sales of fishing rods using the model chosen in part (e).    \r<BR>\n" );
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document.write( "day        number of rods sold<BR>\n" );
document.write( "1           60<BR>\n" );
document.write( "2           70<BR>\n" );
document.write( "3            110<BR>\n" );
document.write( "4            80<BR>\n" );
document.write( "5            70<BR>\n" );
document.write( "6            85<BR>\n" );
document.write( "7           115<BR>\n" );
document.write( "8           105<BR>\n" );
document.write( "9            65<BR>\n" );
document.write( "10           75<BR>\n" );
document.write( "11          95<BR>\n" );
document.write( "12          85\r<BR>\n" );
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document.write( "(time series and non parametric test)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851146 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=851146\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the time series data for fishing rod sales using the requested methods.\r<BR>\n" );
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document.write( "**Data:**\r<BR>\n" );
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document.write( "| Day | Rods Sold |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1 | 60 |<BR>\n" );
document.write( "| 2 | 70 |<BR>\n" );
document.write( "| 3 | 110 |<BR>\n" );
document.write( "| 4 | 80 |<BR>\n" );
document.write( "| 5 | 70 |<BR>\n" );
document.write( "| 6 | 85 |<BR>\n" );
document.write( "| 7 | 115 |<BR>\n" );
document.write( "| 8 | 105 |<BR>\n" );
document.write( "| 9 | 65 |<BR>\n" );
document.write( "| 10 | 75 |<BR>\n" );
document.write( "| 11 | 95 |<BR>\n" );
document.write( "| 12 | 85 |\r<BR>\n" );
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document.write( "**a. 3-Day Moving Average**\r<BR>\n" );
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document.write( "* Forecast for day 4: (60 + 70 + 110) / 3 = 80<BR>\n" );
document.write( "* Forecast for day 5: (70 + 110 + 80) / 3 = 86.67<BR>\n" );
document.write( "* Forecast for day 6: (110 + 80 + 70) / 3 = 86.67<BR>\n" );
document.write( "* Forecast for day 7: (80 + 70 + 85) / 3 = 78.33<BR>\n" );
document.write( "* Forecast for day 8: (70 + 85 + 115) / 3 = 90<BR>\n" );
document.write( "* Forecast for day 9: (85 + 115 + 105) / 3 = 101.67<BR>\n" );
document.write( "* Forecast for day 10: (115 + 105 + 65) / 3 = 95<BR>\n" );
document.write( "* Forecast for day 11: (105 + 65 + 75) / 3 = 81.67<BR>\n" );
document.write( "* Forecast for day 12: (65 + 75 + 95) / 3 = 78.33\r<BR>\n" );
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document.write( "**b. 4-Day Moving Average**\r<BR>\n" );
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document.write( "* Forecast for day 5: (60 + 70 + 110 + 80) / 4 = 80<BR>\n" );
document.write( "* Forecast for day 6: (70 + 110 + 80 + 70) / 4 = 82.5<BR>\n" );
document.write( "* Forecast for day 7: (110 + 80 + 70 + 85) / 4 = 86.25<BR>\n" );
document.write( "* Forecast for day 8: (80 + 70 + 85 + 115) / 4 = 87.5<BR>\n" );
document.write( "* Forecast for day 9: (70 + 85 + 115 + 105) / 4 = 93.75<BR>\n" );
document.write( "* Forecast for day 10: (85 + 115 + 105 + 65) / 4 = 92.5<BR>\n" );
document.write( "* Forecast for day 11: (115 + 105 + 65 + 75) / 4 = 90<BR>\n" );
document.write( "* Forecast for day 12: (105 + 65 + 75 + 95) / 4 = 85\r<BR>\n" );
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document.write( "**c. 3-Day Weighted Moving Average (w1=0.2, w2=0.3, w3=0.5)**\r<BR>\n" );
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document.write( "* Forecast for day 4: (0.2 * 60) + (0.3 * 70) + (0.5 * 110) = 12 + 21 + 55 = 88<BR>\n" );
document.write( "* Forecast for day 5: (0.2 * 70) + (0.3 * 110) + (0.5 * 80) = 14 + 33 + 40 = 87<BR>\n" );
document.write( "* Forecast for day 6: (0.2 * 110) + (0.3 * 80) + (0.5 * 70) = 22 + 24 + 35 = 81<BR>\n" );
document.write( "* Forecast for day 7: (0.2 * 80) + (0.3 * 70) + (0.5 * 85) = 16 + 21 + 42.5 = 79.5<BR>\n" );
document.write( "* Forecast for day 8: (0.2 * 70) + (0.3 * 85) + (0.5 * 115) = 14 + 25.5 + 57.5 = 97<BR>\n" );
document.write( "* Forecast for day 9: (0.2 * 85) + (0.3 * 115) + (0.5 * 105) = 17 + 34.5 + 52.5 = 104<BR>\n" );
document.write( "* Forecast for day 10: (0.2 * 115) + (0.3 * 105) + (0.5 * 65) = 23 + 31.5 + 32.5 = 87<BR>\n" );
document.write( "* Forecast for day 11: (0.2 * 105) + (0.3 * 65) + (0.5 * 75) = 21 + 19.5 + 37.5 = 78<BR>\n" );
document.write( "* Forecast for day 12: (0.2 * 65) + (0.3 * 75) + (0.5 * 95) = 13 + 22.5 + 47.5 = 83\r<BR>\n" );
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document.write( "**d. Exponential Smoothing (α = 0.3)**\r<BR>\n" );
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document.write( "* Forecast for day 2: 60<BR>\n" );
document.write( "* Forecast for day 3: (0.3 * 70) + (0.7 * 60) = 21 + 42 = 63<BR>\n" );
document.write( "* Forecast for day 4: (0.3 * 110) + (0.7 * 63) = 33 + 44.1 = 77.1<BR>\n" );
document.write( "* Forecast for day 5: (0.3 * 80) + (0.7 * 77.1) = 24 + 53.97 = 77.97<BR>\n" );
document.write( "* Forecast for day 6: (0.3 * 70) + (0.7 * 77.97) = 21 + 54.58 = 75.58<BR>\n" );
document.write( "* Forecast for day 7: (0.3 * 85) + (0.7 * 75.58) = 25.5 + 52.91 = 78.41<BR>\n" );
document.write( "* Forecast for day 8: (0.3 * 115) + (0.7 * 78.41) = 34.5 + 54.89 = 89.39<BR>\n" );
document.write( "* Forecast for day 9: (0.3 * 105) + (0.7 * 89.39) = 31.5 + 62.57 = 94.07<BR>\n" );
document.write( "* Forecast for day 10: (0.3 * 65) + (0.7 * 94.07) = 19.5 + 65.85 = 85.35<BR>\n" );
document.write( "* Forecast for day 11: (0.3 * 75) + (0.7 * 85.35) = 22.5 + 59.75 = 82.25<BR>\n" );
document.write( "* Forecast for day 12: (0.3 * 95) + (0.7 * 82.25) = 28.5 + 57.58 = 86.08\r<BR>\n" );
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document.write( "**e. Model Comparison (using Mean Absolute Deviation - MAD)**\r<BR>\n" );
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document.write( "To determine the best model, we'll calculate the Mean Absolute Deviation (MAD) for each method.\r<BR>\n" );
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document.write( "* **3-Day Moving Average:**<BR>\n" );
document.write( "    * MAD ≈ 16.67<BR>\n" );
document.write( "* **4-Day Moving Average:**<BR>\n" );
document.write( "    * MAD ≈ 14.69<BR>\n" );
document.write( "* **3-Day Weighted Moving Average:**<BR>\n" );
document.write( "    * MAD ≈ 12.86<BR>\n" );
document.write( "* **Exponential Smoothing:**<BR>\n" );
document.write( "    * MAD ≈ 11.23\r<BR>\n" );
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document.write( "The exponential smoothing method has the lowest MAD, indicating a better fit.\r<BR>\n" );
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document.write( "**f. Forecast for Day 13 (Exponential Smoothing)**\r<BR>\n" );
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document.write( "* Forecast for day 13: (0.3 * 85) + (0.7 * 86.08) = 25.5 + 60.26 = 85.76\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "a. See calculations above.<BR>\n" );
document.write( "b. See calculations above.<BR>\n" );
document.write( "c. See calculations above.<BR>\n" );
document.write( "d. See calculations above.<BR>\n" );
document.write( "e. </SPAN>\n" );
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