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.
How many individual ratings cannot be determined from the above information? [This question was asked as TITA]

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is A

The problem requires checking if all individual ratings can be uniquely determined using the provided mean ratings of restaurants and workers, along with a few specific ratings.  

Step 1: Total ratings per restaurant
Each restaurant rated 5 workers, so number of ratings per restaurant = 5. 

Step 2: Total sum of ratings for each restaurant
 

• R1: Mean = 3.4 → Total = \( 3.4 \times 5 = 17 \)
• R2: Mean = 2.2 → Total = \( 2.2 \times 5 = 11 \)
• R3: Mean = 3.8 → Total = \( 3.8 \times 5 = 19 \)
• R4: Mean = 2.8 → Total = \( 2.8 \times 5 = 14 \)
• R5: Mean = 3.4 → Total = \( 3.4 \times 5 = 17 \)

Step 3: Use known ratings
 

• R1 gave 5 to Waman → Remaining sum = \( 17 - 5 = 12 \)
• R2 gave 5 to Xavier, 1 to Waman, 1 to Yusuf → Remaining = \( 11 - 7 = 4 \)
• R3 gave 5 to Waman, 5 to Xavier, 1 to Yusuf → Remaining = \( 19 - 11 = 8 \)
• R5 gave 5 to Vasu → Remaining = \( 17 - 5 = 12 \)

Step 4: Worker ratings

Worker	Known Ratings	Mean
Ullas	1 (R1), ?, ?, ?, ?	2.2
Vasu	?, ?, ?, ?, 5 (R5)	3.8
Waman	5 (R1), 1 (R2), 5 (R3), ?, ?	3.4
Xavier	?, 5 (R2), 5 (R3), ?, ?	3.6
Yusuf	?, 1 (R2), 1 (R3), ?, ?	2.6

Step 5: Can all ratings be uniquely determined?
Yes, because:

• The total ratings per restaurant are fixed.
• The individual mean ratings per worker provide equations for the worker’s rating sum.
• There are enough known ratings to solve the system of equations uniquely.

Final Answer: \( \boxed{0} \) ratings cannot be determined.