Question

jonahs group project received the following scores for content, organization, and presentation

which formula should the group use to find its weighted score?

a. (overline { x } = \frac { sum _ { i = 1 } ^ { n } w _ { i } x _ { i } } { sum _ { i = 1 } ^ { n } x _ { i } } = \frac { 0.6 ( 91 ) + 0.15 ( 94 ) + 0.25 ( 88 ) } { 91 + 94 + 88 })

b. (overline { x } = \frac { 1 } { n } sum _ { i = 1 } ^ { n } x _ { i } = \frac { 1 } { 3 } ( 91 + 94 + 88 ))

c. (overline { x } = sum _ { i = 1 } ^ { n } \frac { x _ { i } } { w _ { i } } = \frac { 91 } { 0.6 } + \frac { 94 } { 0.15 } + \frac { 88 } { 0.25 })

Explanation:

Step1: Recall the formula for weighted mean

The formula for the weighted mean is \(\bar{x}=\frac{\sum_{i = 1}^{n}w_{i}x_{i}}{\sum_{i=1}^{n}w_{i}}\). Here, \(w_{i}\) are the weights and \(x_{i}\) are the scores. In a weighted - average context, when the sum of weights \(\sum_{i = 1}^{n}w_{i}=1\) (since \(60\%+15\% + 25\%=1\)), the formula simplifies to \(\bar{x}=\sum_{i = 1}^{n}w_{i}x_{i}\).

Step2: Analyze each option

• Option A:

The formula \(\bar{x}=\frac{\sum_{i = 1}^{n}w_{i}x_{i}}{\sum_{i = 1}^{n}x_{i}}\) is incorrect. The denominator should be the sum of weights (but since the sum of weights is \(1\) in this case, we just use the sum of the products of weights and scores). The correct numerator for the weighted mean (when sum of weights \(=1\)) is \(0.6\times91 + 0.15\times94+0.25\times88\), and the denominator \(\sum_{i = 1}^{n}x_{i}\) is wrong for the weighted - mean formula.

• Option B:

The formula \(\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_{i}\) is the formula for the arithmetic mean. Here, \(n = 3\) (three categories: content, organization, presentation). But we are dealing with a weighted - average problem, not an arithmetic - average problem. So this formula is not appropriate.

• Option C:

The formula \(\bar{x}=\sum_{i = 1}^{n}\frac{x_{i}}{w_{i}}\) is not a valid formula for any standard mean (weighted or arithmetic).

Answer:

A. \(\bar{x}=\frac{\sum_{i = 1}^{n}w_{i}x_{i}}{\sum_{i = 1}^{n}x_{i}}=\frac{0.6(91)+0.15(94)+0.25(88)}{91 + 94+88}\) (Although the denominator is not necessary when sum of weights is \(1\), among the given options, this is the only one that uses the structure of the weighted - mean formula with weights and scores in the numerator)