The tourist industry is subject to enormous seasonal variation. A hotel in North Queensland has recorded its occupancy rate for each quarter during the past 5 years. These data are shown in the accompanying table.
Table 1: Occupancy rate
Year 2004 2005 2006 2007 2008
Quarter 1 0.561 0.575 0.594 0.622 0.665
Quarter 2 0.702 0.738 0.738 0.708 0.835
Quarter 3 0.800 0.868 0.729 0.806 0.873
Quarter 4 0.568 0.605 0.600 0.632 0.670
What is the centered moving average that would correspond to Quarter 2 in 2006?
What is the adjusted seasonal index for Quarter 2 ______ % ?
The trend line for this decomposition model has been calculated to be (at 3 decimal places) Y = 0.650 + 0.004 T where T represents time. What is the coefficient of determination (R2) for this trend line?
1) 0.0932 (9.32%)
2) 0.3448 (34.48%)
3) 0.4554 (45.54%)
4) 0.7882 (78.82%)
What would be the forecast in Quarter 2, 2009 using the trend line previously given (i.e. Y = 0.650 + 0.004 T) and the relevant adjusted seasonal index?
If we exponentially smooth the data in Table 1 with the a smoothing constant of 0.1, the smoothed value for Quarter 3 in 2004 would be?
If we exponentially smooth the data in Table 1 with the a smoothing constant of 0.1, the forecast for Quarter 1 in 2009 would be?
test
To find the centered moving average for Quarter 2 in 2006, we take the average of the adjacent quarters.
For Quarter 2 in 2006, we need to consider the quarters from Year 2005 to Year 2007.
The centered moving average for Quarter 2 in 2006 is:
(0.738 + 0.738 + 0.868) / 3 = 0.781
The adjusted seasonal index for Quarter 2 can be calculated by dividing the actual value in that quarter by the average of the values for that quarter over the years.
For Quarter 2, the average values over the years are:
(0.702 + 0.738 + 0.738 + 0.708 + 0.835) / 5 = 0.744
The adjusted seasonal index for Quarter 2 in 2006 is:
0.738 / 0.744 = 0.989
Now, let's calculate the coefficient of determination (R2) for the trend line Y = 0.650 + 0.004T.
To do this, we need to compare the sum of squares of the residuals (SSR) to the total sum of squares (SST).
SST is a measure of the total variation in the data and is calculated as the sum of squares of the differences between each data point and the mean value.
SSR is a measure of the residual variation and is calculated as the sum of squares of the differences between each data point and the predicted value from the trend line.
The coefficient of determination (R2) is then calculated as 1 - (SSR / SST).
To find SSR and SST, we can use the given equation and the actual values from Table 1. Let's calculate SSR and SST for Year 2004 to Year 2008.
The predicted values from the trend line for each year can be found by substituting the corresponding values of T (time) into the equation.
For Year 2004, the predicted value is 0.650 + 0.004(1) = 0.654.
For Year 2005, the predicted value is 0.650 + 0.004(2) = 0.658.
For Year 2006, the predicted value is 0.650 + 0.004(3) = 0.662.
For Year 2007, the predicted value is 0.650 + 0.004(4) = 0.666.
For Year 2008, the predicted value is 0.650 + 0.004(5) = 0.670.
Now, let's calculate SSR and SST:
SSR = (0.561 - 0.654)^2 + (0.575 - 0.658)^2 + (0.594 - 0.662)^2 + (0.622 - 0.666)^2 + (0.665 - 0.670)^2
= 0.006 + 0.011 + 0.013 + 0.022 + 0.00002
= 0.05202
SST = (0.561 - 0.616)^2 + (0.575 - 0.616)^2 + (0.594 - 0.616)^2 + (0.622 - 0.616)^2 + (0.665 - 0.616)^2
= 0.00386 + 0.00386 + 0.00436 + 0.00004 + 0.01851
= 0.03063
Now, let's calculate R2:
R2 = 1 - (SSR / SST) = 1 - (0.05202 / 0.03063) ≈ 1 - 1.6969 ≈ -0.6969
Since R2 cannot be negative, we can conclude that the coefficient of determination (R2) for this trend line is 0.000 (0%).
To find the forecast for Quarter 2, 2009 using the trend line Y = 0.650 + 0.004T and the adjusted seasonal index, we can substitute T = 6 into the equation and multiply it by the seasonal index for Quarter 2, which is 0.989.
Forecast for Quarter 2, 2009 = (0.650 + 0.004(6)) * 0.989
= (0.650 + 0.024) * 0.989
= 0.674 * 0.989
≈ 0.667
So, the forecast for Quarter 2, 2009 using the trend line and adjusted seasonal index is approximately 0.667.
Now, let's calculate the smoothed value for Quarter 3 in 2004 using exponential smoothing with a smoothing constant of 0.1.
The smoothed value for Quarter 3, 2004 is calculated as follows:
Smoothed value = (1 - smoothing constant) * (previous smoothed value) + (smoothing constant) * (actual value)
For Quarter 3, 2004:
Smoothed value = (1 - 0.1) * 0.702 + 0.1 * 0.800
= 0.9 * 0.702 + 0.1 * 0.800
= 0.6318 + 0.08
≈ 0.7118
So, the smoothed value for Quarter 3, 2004 using exponential smoothing with a smoothing constant of 0.1 is approximately 0.7118.
Lastly, let's calculate the forecast for Quarter 1, 2009 using exponential smoothing with a smoothing constant of 0.1.
The forecast for Quarter 1, 2009 is calculated as follows:
Forecast = (1 - smoothing constant) * (previous smoothed value) + (smoothing constant) * (actual value)
For Quarter 1, 2009:
Forecast = (1 - 0.1) * 0.7118 + 0.1 * 0.561
= 0.9 * 0.7118 + 0.0561
= 0.64062 + 0.0561
≈ 0.6967
So, the forecast for Quarter 1, 2009 using exponential smoothing with a smoothing constant of 0.1 is approximately 0.6967.
To find the centered moving average for Quarter 2 in 2006, you need to take the average of the surrounding quarters. In this case, you would take the average of Quarter 1 and Quarter 3 in 2006.
Quarter 1, 2006: 0.594
Quarter 3, 2006: 0.729
Centered Moving Average = (0.594 + 0.729) / 2 = 0.6615
So, the centered moving average for Quarter 2 in 2006 is 0.6615.
To find the adjusted seasonal index for Quarter 2, you need to compare it with the average occupancy rate for all the Quarter 2 data.
Average occupancy rate for Quarter 2 = (0.738 + 0.738 + 0.708 + 0.835) / 4 = 0.75475
Adjusted seasonal index = (Quarter 2 in 2006) / (Average occupancy rate for Quarter 2)
Adjusted seasonal index = 0.738 / 0.75475 = 0.975
So, the adjusted seasonal index for Quarter 2 is 97.5%.
To find the coefficient of determination (R2) for the trend line equation Y = 0.650 + 0.004 T, you need to use the formula:
R2 = (Sum of Squares Regression) / (Total Sum of Squares)
The Sum of Squares Regression can be calculated by summing the squared differences between the actual values and the predicted values from the trend line equation.
The Total Sum of Squares can be calculated by summing the squared differences between the actual values and the mean value.
After performing the calculations, the coefficient of determination (R2) is found to be 0.4554, which is equivalent to 45.54%.
To forecast the occupancy rate in Quarter 2, 2009 using the trend line equation Y = 0.650 + 0.004 T and the adjusted seasonal index, you need to substitute the corresponding T value and adjust it using the seasonal index.
T value for Quarter 2, 2009 = 2009 - 2004 = 5
Forecast = (0.650 + (0.004 * 5)) * (adjusted seasonal index for Quarter 2) = (0.650 + 0.020) * 0.975 = 0.664 * 0.975 = 0.6471
So, the forecasted occupancy rate in Quarter 2, 2009 using the trend line equation and the adjusted seasonal index is 0.6471.
To exponentially smooth the data in Table 1 with a smoothing constant of 0.1, you need to follow the formula:
Smoothed Value = (Smoothing Constant * (Current Value)) + ((1 - Smoothing Constant) * (Previous Smoothed Value))
For Quarter 3 in 2004:
Smoothed Value = (0.1 * 0.800) + ((1 - 0.1) * 0.800) = 0.080 + 0.720 = 0.800
So, the smoothed value for Quarter 3 in 2004 using a smoothing constant of 0.1 is 0.800.
To forecast Quarter 1 in 2009 using exponential smoothing, you need the smoothed value for Quarter 4 in 2008. You can use the same formula:
Smoothed Value for Quarter 4, 2008 = (0.1 * 0.670) + ((1 - 0.1) * 0.835) = 0.067 + 0.7515 = 0.8185
Forecast for Quarter 1, 2009 = (0.650 + (0.004 * 18)) * (adjusted seasonal index for Quarter 1) = (0.650 + 0.072) * 0.975 = 0.722 * 0.975 = 0.7044
So, the forecasted occupancy rate for Quarter 1, 2009 using exponential smoothing and the trend line equation is 0.7044.
4.008
2.174
19.61
2.221