(a) Calculate the 4-year moving average of the following data related to the sales in a department store.

To calculate the 4-year moving average, we will follow these steps:

1. Break Down the Problem

• Identify the sales data points for each year.
• Determine how to compute the moving average over 4 years.

2. Relevant Concepts

• A moving average is calculated by taking the average of a specific number of data points.
• For a 4-year moving average, sum the sales for a 4-year period and divide by 4.

3. Analysis and Detail

• Suppose the sales data for the years is as follows:
  • Year 1: Sales = S1
  • Year 2: Sales = S2
  • Year 3: Sales = S3
  • Year 4: Sales = S4
  • Year 5: Sales = S5
  • Year 6: Sales = S6
  • Year 7: Sales = S7
  • Year 8: Sales = S8

The formula for the 4-year moving average at year $t$ is given by:

$MA_t = \frac{S_{t-3} + S_{t-2} + S_{t-1} + S_t}{4}$

• The moving averages will be calculated for Year 4 through Year 8.

4. Verify and Summarize

• Calculate the moving averages for the defined years and confirm that they are correctly computed using our formula.

Final Answer

Assuming hypothetical sales data for 8 years as follows: 100, 120, 130, 140, 150, 160, 170, 180

• Moving average for Year 4: $MA_4 = \frac{100 + 120 + 130 + 140}{4} = \frac{490}{4} = 122.5$
• Moving average for Year 5: $MA_5 = \frac{120 + 130 + 140 + 150}{4} = \frac{540}{4} = 135.0$
• Moving average for Year 6: $MA_6 = \frac{130 + 140 + 150 + 160}{4} = \frac{580}{4} = 145.0$
• Moving average for Year 7: $MA_7 = \frac{140 + 150 + 160 + 170}{4} = \frac{620}{4} = 155.0$
• Moving average for Year 8: $MA_8 = \frac{150 + 160 + 170 + 180}{4} = \frac{660}{4} = 165.0$

Thus, the 4-year moving averages are:

• Year 4: 122.5
• Year 5: 135.0
• Year 6: 145.0
• Year 7: 155.0
• Year 8: 165.0