Essentials of business analytics 2nd edition camm solutions manual 1

Essentials of Business Analytics

2nd Edition Camm Solutions Manual

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Chapter 8

Time Series Analysis and Forecasting

Solutions:

1. The following table shows the calculations for parts (a), (b), and (c).

c. MAPE = 159.38/5 = 31.88

d. The forecast for week 7 is

= 14.

2. The following table shows the calculations for parts (a), (b), and (c).

a.

= 13.67/5 = 2.73

b. MSE = 54.31/5 = 10.86

c. MAPE = 105.89/5 = 21.18 d. The

3. The following table shows the measures of forecast error for both methods.

For each measure of forecast accuracy the average of all the historical data provided more accurate forecasts than simply using the most recent value.

MSE = 363/6 = 60.5 The forecast for month 8 is

MSE = 216.72/6 = 36.12

c. The average of all the previous values is better because MSE is smaller.

The data appear to follow a horizontal pattern.

b. Three-week moving average.

MSE = 35.67/3 = 11.89

c. Smoothing constant = 0.2

MSE = 65.15/5 = 13.03

d. The three-week moving average provides a better forecast since it has a smaller MSE.

e. Several values of will yield an MSE smaller than the MSE associated with = 0.2. The table below shows the resulting MSE from several different  that you select.

The value of that yields the minimum MSE is = 0.368, which yields an MSE of 12.06.

The data appear to follow a horizontal pattern.

b.Three-week moving average.

MSE = 110/4 = 27.5.

c. Smoothing constant = 0.2

MSE = 252.87/6 = 42.15 The

d.The three-week moving average provides a better forecast since it has a smaller MSE.

e. Several values of will yield an MSE smaller than the MSE associated with = 0.2. The table below shows the resulting MSE for several different values.

The value of that yields the minimum MSE is = 0.351, which yields an MSE of 39.61.

= 0.351

MSE = 237.69/6 = 39.61428577

7. a. Four and Five -week moving averages.

b The MSE for the four-week and five-week moving averages

For the four-week moving average:

MSE = 77.1875/8 = 9.648

For the five-week moving average:

MSE = 51.84/7 = 7.406

c The MSE for the moving average forecasts are:

Using the MSE as our standard, the best number of weeks of past data to use in the moving average computation is five.

8 a. Exponential smoothing forecasts using α = 0.1:

MSE = 101.78/11 = 9.253

For a smoothing constant of α = 0.2:

MSE = 98.80 / 11 = 8.982

Applying the MSE measure of forecast accuracy, a smoothing constant of α = 0.2 produces a smaller MSE and so is preferred.

b.For a smoothing constant of α = 0 1:

MAE = 28.25 / 11 = 2.568 For a smoothing constant of α = 0 2:

Time Series Analysis and Forecasting

MAE = 28.56 / 11 = 2.596

Applying the MAE measure of forecast accuracy, a smoothing constant of α = 0.1 produces a slightly smaller MAE and so is preferred.

c. For a smoothing constant of α = 0 1:

MAPE = 142.46 / 11 = 12.95

For a smoothing constant of α = 0 2:

MAPE = 147.43 / 11 = 13.40

Applying the MAPE measure of forecast accuracy, a smoothing constant of α = 0.1 produces a smaller MAPE and so is preferred.

b.The more recent data receive the greater weight or importance in determining the forecast. The moving averages method weights the last n data values equally in determining the forecast.

The time series plot indicates a horizontal pattern.

b.

The data appear to follow a horizontal pattern.

b. For the three month moving average:

MSE = 11.11 / 9 = 1.235

For the exponential smoothing forecast for α = 0.2:

MSE = 39.80 / 11 = 3.555

Applying the MSE measure of forecast accuracy, a three-month moving average produces a smaller MSE and so is preferred.

c. Using a three-month moving average, the forecast for next month (t = 13) is

The data appear to follow a horizontal pattern.

= 1.08 / 9 = 0.12 MSE(4-Month) = 1.09 / 8 = 0.14

The MSE for the 3-Month moving average is smaller, so use the 3-Month moving average. c. The forecast for month 13 is

b.

The data appear to follow a horizontal pattern.

= 17,988.52 / 9 = 1998.72

= 0.2) = 27,818.49 / 11 = 2528.95

Based on the above MSE values, the 3-month moving average appears better. However, exponential smoothing was penalized by including month 2 which was difficult for any method to forecast. Using only the errors for months 4 to 12, the MSE for exponential smoothing is:

MSE(= 0.2) = 14,694.49 / 9 = 1632.72

Thus, exponential smoothing was better considering months 4 to 12.

c. Using exponential smoothing with = 0.2,

The data appear to follow a horizontal pattern.

b.Smoothing constant = 0.3.

MSE = 5613.18 / 11 = 510.29

c. The MSE values for exponential smoothing forecasts with several different values of appear below.

The values of that yields the smallest possible MSE is = 0.033, which yields an MSE of 459.693

MSE = 5056.62 / 11 = 459.693

The data appear to follow a horizontal pattern.

b.The MSE values for exponential smoothing forecasts with several different values of appear below.

The value of that yields the minimum MSE is = 0.9102, which yields an MSE of 0.0085 Values of  near 0.9102 will yield similar values for the MSE

b.The value of the MSE will vary depending on the ultimate value of that you select. The resulting MSE values for several different values appear below.

The value of that yields the smallest possible MSE is = 0.467, which yields an MSE of 1.22.

MSE = 1.22

c. The forecast for second quarter 2013 will vary depending on the ultimate value of that you selected in part b. Using an exponential smoothing model with = 0.467, the forecast for second quarter of year 3 = 30.93.

a.

The time series plot shows a linear trend.

b.From the Excel output

the regression estimates for the slope and y-intercept that minimize MSE for this time series are are b0 = 4.7 and b1 = 2.1, which results in the following forecasts, errors, and MSE:

17.

The data are following a downward trend b.From the Excel

the regression estimates for the slope and y-intercept that minimize MSE for this time series are are b0 = 119.714 and b1 = -4.929, which results in the following forecasts, errors, and MSE:

The time series plot shows a linear trend

b. From the Excel output

the regression estimates for the slope and y-intercept that minimize MSE for this time series are b0 = 4.717 and b1 = 1.457, which results in the following forecasts, errors, and MSE:

The data appear to follow a downward trend

b.From the Excel output

the regression estimates for the slope and y-intercept that minimize MSE for this time series are are b0 = 13.8 and b1 = -0.7, which results in the following forecasts, errors, and MSE:

Thus, we predict that SCC will achieve a level less than 5% in year 13 (6 years from now) at 4.7%.

The time series plot shows an upward linear trend

b. From the Excel output

the regression estimates for the slope and y-intercept that minimize MSE for this time series are b0 = 19.993 and b1 = 1.774, which results in the following forecasts, errors, and MSE:

c. The average cost/unit has been increasing by approximately $1.77 per year.

ˆ y = b0 + b1t = 19.993 + 1.774(9) = $35.96.

Dummy Variables Year Quarter Qtr1 Qtr2 Qtr3 yt 1 1 1 0 0 71 1 2 0 1 0 48 1 3 0 0 1 58 1 4 0 0 0 78 2 1 1 0 0 68 2 2 0 1 0 41 2 3 0 0 1 60 2 4 0 0 0 81 3 1 1 0 0 62 0 10 20 30 40 50 60 70 80 90 1 2 3 4 5 6 7 8 9 10 11 12 Time Series Value Period (t)

The time series plot shows a horizontal pattern. But, there is a seasonal pattern in the data. For instance, in each year the lowest value occurs in quarter 2 and the highest value occurs in quarter 4.

we can use the Excel Regression tool to find the regression model that to accounts for seasonal effects in the data. From the Excel output

the regression model that minimizes MSE for this time series is: Value

c. The quarterly forecasts for next year are as follows:

Careful scrutiny of the time series plot reveals a horizontal pattern (i.e., a linear trend) with seasonality. For instance, in each year the value drops from quarter 1 to quarter 2 and the value increases from quarter 3 to quarter 4.

b.After putting the data into the following format:

We can use the Excel Regression tool to find the regression model that to accounts for trend and seasonal effects in the data. From the Excel output

The regression model that minimizes MSE for this time series is:

c. Based on the model in part (b), the quarterly forecasts for next year are as follows:

d.After putting the data into the following format:

We can use the Excel Regression tool to find the regression model that to accounts for trend and seasonal effects in the data From the Excel output

The regression model that minimizes MSE for this time series is:

e. Based on the model in part (d), the quarterly forecasts for next year are as follows:

f For the model from part (b) that only includes seasonal effects:

MSE = 335.3333 / 12 = 27.9444

For the model from part (d) that includes both trend and seasonal effects:

The mean squared error for the model from part (d) that includes both trend and seasonal effects is much smaller than the mean squared error for the model from part (b) that includes only seasonal effects. This supports our preliminary conclusions reached in review of the time series plot constructed in part (a) – these data show a linear trend with seasonality.

There appears to be a seasonal pattern in the data and perhaps a moderate upward linear trend.

b.After putting the data into the following format:

We can use the Excel Regression tool to find the regression model that to accounts for seasonal effects in the data From the Excel output

The regression model that minimizes MSE for this time series is:

c. Based on the model in part (b), the quarterly forecasts for next year are as follows:

d.After putting the data into the following format:

we can use the Excel Regression tool to find the regression model that to accounts for seasonal effects in the data From the Excel output

the regression model that minimizes MSE for this time series is:

e Based on the model in part (c), the quarterly forecasts for next year are as follows:

f For the model from part (b) that only includes seasonal effects:

MSE = 124,933.3333 / 12 = 10,411.1111

For the model from part (d) that includes both trend and seasonal effects:

MSE = 56,483.3333 / 12 = 4,706.9444

The mean squared error for the model from part (d) that includes both trend and seasonal effects is much smaller than the mean squared error for the model from part (b) that includes only seasonal effects. This supports our preliminary conclusions reached in review of the time series plot constructed in part (a) – these data show a linear trend with seasonality.

Level of Nitrogen Dioxide

There appears to be a seasonal pattern in the data and perhaps a slight upward linear trend.

b.After putting the data into the following format:

We can use the Excel Regression tool to find the regression model that accounts for the seasonal effects in the data From the Excel output

The regression model that minimizes MSE for this time series is:

Value = 21.6667 + 7.6667HOUR1 + 11.6667HOUR2 + 16.6667HOUR3 + 34.3333HOUR4 +

42.3333HOUR5 + 45HOUR6 + 28.3333HOUR7 + 18.3333HOUR8 + 13.3333HOUR9 + 3.3333HOUR10 + 1.6667HOUR11

c. Using the model estimated in part (b), the hourly forecasts for July 18 (t = 37 through t = 48) are as follows:

d.After putting the data into the following format:

We can use the Excel Regression tool to find the regression model that accounts for the trend and seasonal effects in the data. From the Excel output

The regression model that minimizes MSE for this time series is:

e Using the model estimated in part (d), the hourly forecasts for July 18 (t = 37 through t = 48) are as follows:

f For the model from part (b) that only includes seasonal effects:

For the model from part (d) that includes both trend and seasonal effects:

The mean squared error for the model from part (d) that includes both trend and seasonal effects is somewhat smaller than the mean squared error for the model from part (b) that includes only seasonal effects. This supports our preliminary conclusions reached in review of the time series plot constructed in part (a) – these data show a linear trend with seasonality.