# Question #94b63

Jun 27, 2016

See important assumption below.
The mean attendance of any of two classes by a combined set of students of both groups is, approximately, $60.34$.

#### Explanation:

ASSUMPTION:
These two groups contain different students, that is, there are no students that belong to both groups and attend both classes, so the total number of students is $120 + 150 = 270$, and our task is to evaluate the mean attendance of any of two classes by a combined set of students of both groups.

Let ${x}_{i}$ represent attendance of the first class (XI-A) by an ${i}^{t h}$ person of the first group of ${N}_{1} = 120$ students. It is equal to $1$ if this person attended the first class and $0$ otherwise.
Since the mean attendance in this group equals to ${M}_{1} = 56.35$, we can write the equation:
${M}_{1} = \frac{\Sigma {x}_{i}}{N} _ 1 = \frac{{x}_{1} + {x}_{2} + \ldots + {x}_{120}}{120} = 56.35$
or
$\Sigma {x}_{i} = {x}_{1} + {x}_{2} + \ldots + {x}_{120} = {N}_{1} \cdot {M}_{1} = 120 \cdot 56.35 = 6762$

Let ${y}_{j}$ represent attendance of the second class (XI-B) by an ${j}^{t h}$ person of the second group of ${N}_{2} = 150$ students. It is equal to $1$ if this person attended the second class and $0$ otherwise.
Since the mean attendance in this group equals to ${M}_{2} = 63.45$, we can write the equation:
${M}_{2} = \frac{\Sigma {y}_{j}}{N} _ 2 = \frac{{y}_{1} + {y}_{2} + \ldots + {y}_{150}}{150} = 63.45$
or
$\Sigma {y}_{j} = {y}_{1} + {x}_{2} + \ldots + {y}_{150} = {N}_{2} \cdot {M}_{2} = 150 \cdot 63.45 = 9517.5$

Combining both groups, we have ${N}_{1} + {N}_{2} = 120 + 150 = 270$ students, whose attendance of one of two classes is represented by variables ${x}_{i}$ (where $i \in \left[1 , 120\right]$) and ${y}_{j}$ (where $j \in \left[1 , 150\right]$).
Their mean attendance of any of two classes can be represented as
$M = \frac{\Sigma {x}_{i} + \Sigma {y}_{j}}{{N}_{1} + {N}_{2}} =$
$= \frac{{N}_{1} \cdot {M}_{1} + {N}_{2} \cdot {M}_{2}}{{N}_{1} + {N}_{2}} =$
$= \frac{6762 + 9517.5}{270} \approx 60.34$