Question

1a. write the sigma notation mean formula for the three consecutive - month period that would have the highest mean of the year.

Explanation:

Step1: Recall the mean formula

The mean of a set of values $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i$. For a three - consecutive - month period, let the values for the months be $x_i,x_{i + 1},x_{i+2}$. Here $n = 3$.

Step2: Write the sigma - notation formula

The mean $\bar{x}$ of these three values is $\bar{x}=\frac{1}{3}\sum_{j=0}^{2}x_{i + j}$, where $i$ can take values from 1 to 10 (since we want three - consecutive - month periods within a 12 - month year. When $i = 1$, we have the first three - month period $x_1,x_2,x_3$; when $i=2$, we have $x_2,x_3,x_4$ and so on until $i = 10$ which gives $x_{10},x_{11},x_{12}$). To find the three - consecutive - month period with the highest mean, we would need to calculate $\frac{1}{3}\sum_{j = 0}^{2}x_{i + j}$ for $i=1,2,\cdots,10$ and compare the results.

Answer:

$\bar{x}=\frac{1}{3}\sum_{j=0}^{2}x_{i + j}$ (where $i$ ranges from 1 to 10)