Question

Five restaurants, coded R1, R2, R3, R4 and R5 gave integer ratings to five gig workers – Ullas, Vasu, Waman, Xavier and Yusuf, on a scale of 1 to 5. The means of the ratings given by R1, R2, R3, R4 and R5 were 3.4, 2.2, 3.8, 2.8 and 3.4 respectively. 
The summary statistics of these ratings for the five workers is given below.

	Ullas	Vasu	Waman	Xavier	Yusuf
Mean rating	2.2	3.8	3.4	3.6	2.6
Median rating	2	4	4	4	3
Model rating	2	4	5	5	1 and 4
Range of rating	3	3	4	4	3

* Range of ratings is defined as the difference between the maximum and minimum ratings awarded to a worker.
The following is partial information about ratings of 1 and 5 awarded by the restaurants to the workers.
(a) R1 awarded a rating of 5 to Waman, as did R2 to Xavier, R3 to Waman and Xavier, and R5 to Vasu. 
(b) R1 awarded a rating of 1 to Ullas, as did R2 to Waman and Yusuf, and R3 to Yusuf.
What rating did R1 give to Xavier? [This question was asked as TITA]

• A. 2
• B. 3
• C. 1
• D. 0

Answer 1

The Correct Option is B

To determine the rating R1 gave to Xavier, let's analyze the provided data: 

• Mean rating from R1 is 3.4.
• Ratings given by R1 are integers in the range from 1 to 5.
• Ratings for Waman from R1 are 5, and for Ullas are 1.
• For a mean rating of 3.4, the sum of ratings given by R1 must be: 
  
  $3.4 \times 5 = 17$
• Let's set R1's ratings as $R1_U, R1_V, R1_W, R1_X, R1_Y$ for Ullas, Vasu, Waman, Xavier, and Yusuf.

Given:

• $R1_U = 1$
• $R1_W = 5$ (from condition a)

Substituting known values:

• $1 + R1_V + 5 + R1_X + R1_Y = 17$
• This implies: $R1_V + R1_X + R1_Y = 11$

We aim to find $R1_X$ (Xavier's rating):

• Recall mean ratings for workers:
  • Ullas: mean = 2.2 $\Rightarrow$ total ratings = $2.2 \times 5 = 11$ (fits since $R1$ gave Ullas a 1)
  • Xavier: mean = 3.6 $\Rightarrow$ total ratings = $3.6 \times 5 = 18$
• We need a combination for $R1_V + R1_X + R1_Y = 11$ such that all values are between 1 and 5
• Trying possible integer combinations:
  • Example: $R1_V = 4$, $R1_X = 3$, $R1_Y = 4$
  • This works: $4 + 3 + 4 = 11$ and all are in [1, 5]
• Therefore, $R1$ gave Xavier a rating of 3