Question

1. use summation notation to express the following calculations. a. multiply scores x and y and then add each product. b. sum the scores x and sum the scores y and then multiply the sums. c. subtract x from y and sum the differences. d. sum the x scores.

Explanation:

Step1: Define the number of data - points

Let there be $n$ data - points. The scores are denoted as $X_1,X_2,\cdots,X_n$ and $Y_1,Y_2,\cdots,Y_n$.

Step2: Solution for part a

We multiply each pair of $X_i$ and $Y_i$ and then sum them. The summation notation is $\sum_{i = 1}^{n}X_iY_i$.

Step3: Solution for part b

First, we sum the $X$ scores: $\sum_{i = 1}^{n}X_i$, and sum the $Y$ scores: $\sum_{i = 1}^{n}Y_i$. Then we multiply the sums, so the expression is $(\sum_{i = 1}^{n}X_i)(\sum_{i = 1}^{n}Y_i)$.

Step4: Solution for part c

We subtract $X_i$ from $Y_i$ for each $i$ and sum the differences. The expression is $\sum_{i = 1}^{n}(Y_i - X_i)$.

Step5: Solution for part d

The sum of the $X$ scores is simply $\sum_{i = 1}^{n}X_i$.

Answer:

a. $\sum_{i = 1}^{n}X_iY_i$
b. $(\sum_{i = 1}^{n}X_i)(\sum_{i = 1}^{n}Y_i)$
c. $\sum_{i = 1}^{n}(Y_i - X_i)$
d. $\sum_{i = 1}^{n}X_i$