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Problem 10

# Complete the given table with the values of the 3 -unit moving average of the given function. HINT [See Example 3.] $$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \boldsymbol{s}(\boldsymbol{x}) & 2 & 9 & 7 & 3 & 2 & 5 & 7 & 1 \\ \hline \bar{s}(\boldsymbol{x}) & & & & & & & & \\ \hline \end{array}$$

Expert verified
The 3-unit moving averages for the given function s(x) are: $$\begin{array}{|c|c|} \hline \boldsymbol{x} & \bar{s}(\boldsymbol{x}) \\ \hline 1 & 6 \\ \hline 2 & 6.333 \\ \hline 3 & 4 \\ \hline 4 & 3.333 \\ \hline 5 & 4.667 \\ \hline 6 & 4.333 \\ \hline \end{array}$$
See the step by step solution

## Step 1: Understanding the 3-unit moving average association with given function

Observe that the given function is s(x) with x values ranging from 0 to 7. The idea is to calculate the 3-unit moving average, which will provide the average of every three consecutive data points from the s(x) function.

## Step 2: Calculating the first 3-unit moving average for s(x)

To calculate the 3-unit moving average for the first three data points of s(x), we take the average of the values at x = 0, 1, and 2. In other words, we take the average of s(0), s(1), and s(2): $$\bar{s}(1) = \frac{s(0) + s(1) + s(2)}{3} = \frac{2 + 9 + 7}{3} = \frac{18}{3} = 6$$ The first 3-unit moving average is 6, and it corresponds to the value of x = 1 in the table.

## Step 3: Calculating the remaining 3-unit moving averages for s(x)

Now follow the same procedure for the remaining x values: $$\bar{s}(2) = \frac{s(1) + s(2) + s(3)}{3} = \frac{9 + 7 + 3}{3} = \frac{19}{3} = 6.333$$ $$\bar{s}(3) = \frac{s(2) + s(3) + s(4)}{3} = \frac{7 + 3 + 2}{3} = \frac{12}{3} = 4$$ $$\bar{s}(4) = \frac{s(3) + s(4) + s(5)}{3} = \frac{3 + 2 + 5}{3} = \frac{10}{3} = 3.333$$ $$\bar{s}(5) = \frac{s(4) + s(5) + s(6)}{3} = \frac{2 + 5 + 7}{3} = \frac{14}{3} = 4.667$$ $$\bar{s}(6) = \frac{s(5) + s(6) + s(7)}{3} = \frac{5 + 7 + 1}{3} = \frac{13}{3} = 4.333$$

## Step 4: Inserting the calculated values into the table

Now that we have calculated the 3-unit moving averages, let's insert them into the table: $$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \boldsymbol{s}(\boldsymbol{x}) & 2 & 9 & 7 & 3 & 2 & 5 & 7 & 1 \\ \hline \bar{s}(\boldsymbol{x}) & & 6 & 6.333 & 4 & 3.333 & 4.667 & 4.333 & \\ \hline \end{array}$$ The completed table now shows the 3-unit moving averages for the given function s(x).

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