The data given below for 15 movies on their performance on 3 attribute

Now is the time to provide solution to this question. Before you look at the solution, I do recommend that you first read my article on weighted average what-you-choose-is-what-you-weigh-143895.html. This will make sure, you absorb the concept, and its application better.

Question 1-

In the calculation of Total Score Points, equal weight-age was given to both Performance Score Points and Script Score Points.

Understand the question

3 attributes -Performance, Entertainment, and Script's score points are primary data, and Total score points is a derived data. It means that Total score points is derived out of a mathematical relationship among Performance, Entertainment, and Script score points. The first part of the question i.e. Q1, states that their relationship is governed by weights, or we can simply infer that Total score points is not a simple average of Performance, Entertainment, and Script score points.

Performance, Entertainment, and Script score points might have been mixed in some unknown proportion to derive the value of total score points.

Per the first part of question, Is Weight of Performance score points = Weight of Script score points ?

Traditional Approach

Step 1- Say Total weight = 1. Out of which, weight of Performance score points = weight of Script score points = x, & Weight of Entertainment score points = y. So x+y+x=1 => y=1-2x.

Step 2- Formulate mathematical formula

Total score points = x* Performance score points + y* Entertainment score points + x* Script score points

We can simplify this by plugging in the value of y as 1-2x. So the equation becomes Total score points = x*( Performance score points + Script score points )+ (1-2x)* Entertainment score points.

This gives, \(x=[\frac{(Total score points - Entertainment score points)}{( Performance score points + Script score points - 2 Entertainment score points)}]\)

Step 3- Pick any of the 2 listed movies, and plug in the relevant score points in the equation above. If value of x is same for both the movies, then we an say our assumption of taking weight of Performance score points = weight of Script score points was correct.

Say, we take Predator, & Total Recall movies...

For Predator, \(x= \frac{(5.74-4.7)}{(7.4+4.6-2*4.7)}\) This gives x=0.4.

Now, we try with Total Recall, \(x= \frac{(8.72-8.8)}{(9+8.4-2*8.8)}\) This also gives x=0.4.

This means our assumption of taking weight of Performance score points = weight of Script score points was correct. So the answer is Yes.

Well, I do not really recommend this approach, as this approach involves handling 2 variables x, & y.

Alternate Approach

Step 1- Say weight of Performance score points = weight of Script score points = 1, & weight of Entertainment score points = z. So total weight = 2+z.

Step 2- Formulate mathematical formula

(2+z)* Total score points = 1* Performance score points + z* Entertainment score points + 1* Script score points

This gives, \(z=[\frac{( Performance score points + Entertainment score points - 2* Total score points)}{(Total score points -Script score points)}]\)

Step 3- Pick any of the 2 listed movies, and plug in the relevant score points in the equation above. If value of z is same for both the movies, then we an say our assumption of taking weight of Performance score points = weight of Script score points was correct.

Say, we take Predator, & Total Recall movies...

For Predator, \(z= \frac{(7.4+4.7-2*5.74)}{(5.74-4.6)}\) This gives z=0.5.

Now, we try with Total Recall, \(z= \frac{(9+8.8-2*8.72)}{(8.72-8.4)}\) This also gives z=0.5.

This means our assumption of taking weight of Performance score points = weight of Script score points was correct. So the answer is Yes.

This means that Performance, Entertainment, and Script score points are weighted in the ratio of 1:0.5:1 => 2:1:2 => 40%:20%:40% (In terms of %)

Question 2-

If the weight-age of Entertainment Score Points is doubled to calculate Total Score Points, then there are exactly 9 movies, whose Total Score Points will increase.

Understand the question

Presently, the weight-age of Entertainment Score Points is 20% out of 100%. Now it is doubled. It means that it becomes 40% now. Not to accommodate extra 20% (40-20), Performance, & Script score points will have to sacrifice 10% each. So the new weigh-age of 3 attribute would be 30%:40%:30%.

Approach

Before answering this question, let us look at the concept of weighted average. If you have not visited my weighted article till now, pl. do so now.what-you-choose-is-what-you-weigh-143895.html

Say, there are 3 numbers 10, 12 & 20. Their average would be 14.
Now, a one more number 12 is added to it. So we have 4 #. 10, 12, 12 & 20.

Now, my question to you is- Can you deduce whether the average of 4 # would be less than 14 or more than 14?

Weighted Average Fundamental- If a new data, whose value is less current average of few data is added, then new average would be less than the current average. Similarly, vise-versa true for greater value.

Why is so?- It is because, the the new data pulls the average towards it.

Now, coming to our question- If the weight-age of Entertainment Score Points is doubled to calculate Total Score Points, then there are exactly 9 movies, whose Total Score Points will increase.

We are interested to know, new Total Score Points will increase compared to current Total Score Points or not.

So per Weighted Average Fundamental , Entertainment Score Points weight-age is doubled, or we can say that a new data is included. We can deduce that for those movies, whose Entertainment Score Points is more than Total Score Points, new Total Score Points will increase.

There are exactly 9 movies, whose Entertainment Score Points is more than Total Score Points. these are Lara Croft, Rocky, Rocky 3, Rocky Balboa, Salt, Tango & Cash, Terminator, Total Recall, True Lies. Answer is Yes.

Question 3-

If for each of the Arnold Swaznegger movies, respective movie Performance Score Point is swapped with Script Score Point, then the Total Score Point of exactly one movie of Arnold Swaznegger will decrease.

Understand the question

Currently, Performance, Entertainment, and Script score points are weighed in the ratio of 40%:20%:40%. The weights of Performance, and Script score points are equal, hence swapping will have no impact. Answer is N0.

Hope this explanation helps.

-Shalabh

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