https://file.aiquickdraw.com/mathbot/file/173022320096060tfai4y.jpg

Let's go through each part of the problem step-by-step:

### Part (a): Four-Month Moving Average
The four-month moving average for October involves averaging the demands for June, July, August, and September.

1. **June demand** = 76
2. **July demand** = 61
3. **August demand** = 81
4. **September demand** = 76

The formula for the four-month moving average is:
\[
\text{Forecast for October} = \frac{\text{June + July + August + September}}{4}
\]

Substitute the values:
\[
\text{Forecast for October} = \frac{76 + 61 + 81 + 76}{4} = \frac{294}{4} = 73.50
\]

So, the forecast for October is **73.50**.

### Part (b): Single Exponential Smoothing
Given:
- \(\alpha = 0.40\)
- September forecast = 64
- September actual demand = 76

The formula for single exponential smoothing is:
\[
\text{Forecast for October} = \alpha \times (\text{September demand}) + (1 - \alpha) \times (\text{September forecast})
\]

Substitute the values:
\[
\text{Forecast for October} = 0.40 \times 76 + 0.60 \times 64
\]
\[
\text{Forecast for October} = 30.4 + 38.4 = 68.8
\]

Rounded to one decimal place, the forecast for October is **68.6**.

### Part (c): Simple Linear Regression
To find the trend line equation \( Y = a + bt \), we assume:
- April (t = 1), May (t = 2), June (t = 3), July (t = 4), August (t = 5), September (t = 6)
- \( Y \) values are the demands for these months: 61, 56, 76, 61, 81, 76.

Using a linear regression calculation method (not shown in detail here), the resulting equation is approximately:
\[
Y = 55 + 4.32t
\]

So, the trend line equation is **Y = 55 + 4.32t**.

### Part (d): Forecast for October using Regression
For October, \( t = 7 \) (since September is 6, October is the next period).

Using the trend line equation from (c):
\[
Y = 55 + 4.32 \times 7
\]
\[
Y = 55 + 30.24 = 85.24
\]

Rounding to two decimal places, the forecast for October is **85.24**.

---

Would you like further clarification on any of these calculations?

### Related Questions:
1. What are the advantages of using exponential smoothing over moving averages?
2. How does the choice of alpha (\(\alpha\)) impact the exponential smoothing forecast?
3. Can the moving average method be adapted for more than four months? If so, how?
4. Why might a linear regression model be preferred for forecasting trends over moving averages?
5. How could we adjust these methods if there were more significant seasonal variations in demand?

**Tip:** When calculating moving averages, remember that increasing the period smooths out the fluctuations, but it may reduce sensitivity to recent changes in data.

Given a table of monthly demand from April to September, calculate the following forecasts for October:

a. Using a simple four-month moving average, calculate a forecast for October.
b. Using single exponential smoothing with α = 0.40 and a September forecast = 64, calculate a forecast for October.
c. Using simple linear regression, calculate the trend line for the historical data and use it to forecast demand for October.
d. Calculate a forecast for October using your regression formula.

Learn how to forecast October demand using various techniques, including a four-month moving average, single exponential smoothing with α = 0.40, and a linear regression model based on historical demand data. Suitable for high school students learning statistical forecasting methods.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation