CAT 2023 Slot 1 DILR Question & Solution
Data Set
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.
* 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.
Question 1
How many individual ratings cannot be determined from the above information?
Solution:
Given that 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 sum of ratings given by R1, R2, R3 R4, R5 are 5*means = 17, 11, 19, 14, and 17 respectively.
Similarly the sum of ratings received by U, V, W, X and Y are 5*means = 11, 19, 17, 18, and 13 respectively.
Also capturing the absolute data given in the partial information (a) and (b) and representing as a table, we get:
Now,
Consider U
Given median = 2, mode = 2 and range = 3
=> His ratings should be of the form 1, a , 2, b, 4 => 1 + 2 + 4 + a + b = 11 => a + b = 4. For mode = 2 => a = b = 2
=> U's ratings are 1, 2, 2, 2, 4.
Consider V
Given median = 4, mode = 4 and range = 3
=> His ratings should be of the form 2, a, 4, b, 5 => 2 + 4 + 5 + a + b = 19 => a + b = 8 => For mode = 4 => a = b = 4
=> V's ratings are 2, 4, 4, 4, 5.
Consider W
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => 1 + a + 4 + 5 + 5 = 17 => a = 2
=> W's ratings are 1, 2, 4, 5, 5.
Consider X
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => a + 1 + 4 + 5 + 5 = 18 => a = 3
=> X's ratings are 1, 3, 4, 5, 5
Consider Y
Given median = 3, mode = 1 & 4, Range = 3
=> His ratings are 1, 1, 3, 4, 4.
Capturing this data in the table, we get:
Now, consider column R3 => The two missing entries should add up to 19 - 1 - 5 - 5 = 8, (only possibility is 4 + 4) => We can fill the row "U" and 4 in the row "V"
Now, consider column R2 => Missing entry should be 11 - 2 - 1 - 5 - 1 = 2
Consider column R1, the missing elements should add up to 17 - 5 - 4 - 1 = 7 (3 + 4 or 4 + 3) ----(1)
Consider R5, the missing elements should add up to 10 => 2 + 4 + 4 or 4 + 3 + 3 (not possible) as (1) requires a 3.
Now, we can fill column R1 as 3 + 4 and the remaining in column R4 and we can get the complete table
=> All ratings can be determined uniquely => 0.
Question 2
To how many workers did R2 give a rating of 4?
Solution:
Given that 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 sum of ratings given by R1, R2, R3 R4, R5 are 5*means = 17, 11, 19, 14, and 17 respectively.
Similarly the sum of ratings received by U, V, W, X and Y are 5*means = 11, 19, 17, 18, and 13 respectively.
Also capturing the absolute data given in the partial information (a) and (b) and representing as a table, we get:
Now,
Consider U
Given median = 2, mode = 2 and range = 3
=> His ratings should be of the form 1, a , 2, b, 4 => 1 + 2 + 4 + a + b = 11 => a + b = 4. For mode = 2 => a = b = 2
=> U's ratings are 1, 2, 2, 2, 4.
Consider V
Given median = 4, mode = 4 and range = 3
=> His ratings should be of the form 2, a, 4, b, 5 => 2 + 4 + 5 + a + b = 19 => a + b = 8 => For mode = 4 => a = b = 4
=> V's ratings are 2, 4, 4, 4, 5.
Consider W
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => 1 + a + 4 + 5 + 5 = 17 => a = 2
=> W's ratings are 1, 2, 4, 5, 5.
Consider X
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => a + 1 + 4 + 5 + 5 = 18 => a = 3
=> X's ratings are 1, 3, 4, 5, 5
Consider Y
Given median = 3, mode = 1 & 4, Range = 3
=> His ratings are 1, 1, 3, 4, 4.
Capturing this data in the table, we get:
Now, consider column R3 => The two missing entries should add up to 19 - 1 - 5 - 5 = 8, (only possibility is 4 + 4) => We can fill the row "U" and 4 in the row "V"
Now, consider column R2 => Missing entry should be 11 - 2 - 1 - 5 - 1 = 2
Consider column R1, the missing elements should add up to 17 - 5 - 4 - 1 = 7 (3 + 4 or 4 + 3) ----(1)
Consider R5, the missing elements should add up to 10 => 2 + 4 + 4 or 4 + 3 + 3 (not possible) as (1) requires a 3.
Now, we can fill column R1 as 3 + 4 and the remaining in column R4 and we can get the complete table
R2 gave ratings of 1, 1, 2, 2, 5 => He gave 4 to 0 workers => 0 is the answer.
Question 3
What rating did R1 give to Xavier?
Solution:
Given that 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 sum of ratings given by R1, R2, R3 R4, R5 are 5*means = 17, 11, 19, 14, and 17 respectively.
Similarly the sum of ratings received by U, V, W, X and Y are 5*means = 11, 19, 17, 18, and 13 respectively.
Also capturing the absolute data given in the partial information (a) and (b) and representing as a table, we get:
Now,
Consider U
Given median = 2, mode = 2 and range = 3
=> His ratings should be of the form 1, a , 2, b, 4 => 1 + 2 + 4 + a + b = 11 => a + b = 4. For mode = 2 => a = b = 2
=> U's ratings are 1, 2, 2, 2, 4.
Consider V
Given median = 4, mode = 4 and range = 3
=> His ratings should be of the form 2, a, 4, b, 5 => 2 + 4 + 5 + a + b = 19 => a + b = 8 => For mode = 4 => a = b = 4
=> V's ratings are 2, 4, 4, 4, 5.
Consider W
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => 1 + a + 4 + 5 + 5 = 17 => a = 2
=> W's ratings are 1, 2, 4, 5, 5.
Consider X
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => a + 1 + 4 + 5 + 5 = 18 => a = 3
=> X's ratings are 1, 3, 4, 5, 5
Consider Y
Given median = 3, mode = 1 & 4, Range = 3
=> His ratings are 1, 1, 3, 4, 4.
Capturing this data in the table, we get:
Now, consider column R3 => The two missing entries should add up to 19 - 1 - 5 - 5 = 8, (only possibility is 4 + 4) => We can fill the row "U" and 4 in the row "V"
Now, consider column R2 => Missing entry should be 11 - 2 - 1 - 5 - 1 = 2
Consider column R1, the missing elements should add up to 17 - 5 - 4 - 1 = 7 (3 + 4 or 4 + 3) ----(1)
Consider R5, the missing elements should add up to 10 => 2 + 4 + 4 or 4 + 3 + 3 (not possible) as (1) requires a 3.
Now, we can fill column R1 as 3 + 4 and the remaining in column R4 and we can get the complete table
=> From the table, we can see that R1 gave a rating of 3 to Xavier.
Question 4
What is the median of the ratings given by R3 to the five workers?
Solution:
Given that 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 sum of ratings given by R1, R2, R3 R4, R5 are 5*means = 17, 11, 19, 14, and 17 respectively.
Similarly the sum of ratings received by U, V, W, X and Y are 5*means = 11, 19, 17, 18, and 13 respectively.
Also capturing the absolute data given in the partial information (a) and (b) and representing as a table, we get:
Now,
Consider U
Given median = 2, mode = 2 and range = 3
=> His ratings should be of the form 1, a , 2, b, 4 => 1 + 2 + 4 + a + b = 11 => a + b = 4. For mode = 2 => a = b = 2
=> U's ratings are 1, 2, 2, 2, 4.
Consider V
Given median = 4, mode = 4 and range = 3
=> His ratings should be of the form 2, a, 4, b, 5 => 2 + 4 + 5 + a + b = 19 => a + b = 8 => For mode = 4 => a = b = 4
=> V's ratings are 2, 4, 4, 4, 5.
Consider W
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => 1 + a + 4 + 5 + 5 = 17 => a = 2
=> W's ratings are 1, 2, 4, 5, 5.
Consider X
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => a + 1 + 4 + 5 + 5 = 18 => a = 3
=> X's ratings are 1, 3, 4, 5, 5
Consider Y
Given median = 3, mode = 1 & 4, Range = 3
=> His ratings are 1, 1, 3, 4, 4.
Capturing this data in the table, we get:
Now, consider column R3 => The two missing entries should add up to 19 - 1 - 5 - 5 = 8, (only possibility is 4 + 4) => We can fill the row "U" and 4 in the row "V"
Now, consider column R2 => Missing entry should be 11 - 2 - 1 - 5 - 1 = 2
Consider column R1, the missing elements should add up to 17 - 5 - 4 - 1 = 7 (3 + 4 or 4 + 3) ----(1)
Consider R5, the missing elements should add up to 10 => 2 + 4 + 4 or 4 + 3 + 3 (not possible) as (1) requires a 3.
Now, we can fill column R1 as 3 + 4 and the remaining in column R4 and we can get the complete table
=> Ratings give by R3 are 1, 4, 4, 5, 5 => Median = 4.
Question 5
Which among the following restaurants gave its median rating to exactly one of the workers?
Solution:
Given that 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 sum of ratings given by R1, R2, R3 R4, R5 are 5*means = 17, 11, 19, 14, and 17 respectively.
Similarly the sum of ratings received by U, V, W, X and Y are 5*means = 11, 19, 17, 18, and 13 respectively.
Also capturing the absolute data given in the partial information (a) and (b) and representing as a table, we get:
Now,
Consider U
Given median = 2, mode = 2 and range = 3
=> His ratings should be of the form 1, a , 2, b, 4 => 1 + 2 + 4 + a + b = 11 => a + b = 4. For mode = 2 => a = b = 2
=> U's ratings are 1, 2, 2, 2, 4.
Consider V
Given median = 4, mode = 4 and range = 3
=> His ratings should be of the form 2, a, 4, b, 5 => 2 + 4 + 5 + a + b = 19 => a + b = 8 => For mode = 4 => a = b = 4
=> V's ratings are 2, 4, 4, 4, 5.
Consider W
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => 1 + a + 4 + 5 + 5 = 17 => a = 2
=> W's ratings are 1, 2, 4, 5, 5.
Consider X
Given median = 4, mode = 5 and range = 4
=> His ratings should be of the form 1, a, 4, 5, 5 => a + 1 + 4 + 5 + 5 = 18 => a = 3
=> X's ratings are 1, 3, 4, 5, 5
Consider Y
Given median = 3, mode = 1 & 4, Range = 3
=> His ratings are 1, 1, 3, 4, 4.
Capturing this data in the table, we get:
Now, consider column R3 => The two missing entries should add up to 19 - 1 - 5 - 5 = 8, (only possibility is 4 + 4) => We can fill the row "U" and 4 in the row "V"
Now, consider column R2 => Missing entry should be 11 - 2 - 1 - 5 - 1 = 2
Consider column R1, the missing elements should add up to 17 - 5 - 4 - 1 = 7 (3 + 4 or 4 + 3) ----(1)
Consider R5, the missing elements should add up to 10 => 2 + 4 + 4 or 4 + 3 + 3 (not possible) as (1) requires a 3.
Now, we can fill column R1 as 3 + 4 and the remaining in column R4 and we can get the complete table
=> R2 median rating is 2 => given to 2 workers
=> R5 median rating is 4 => given to 2 workers
=> R4 median rating is 3 => given to only 1 worker.
=> R3 median rating is 4 => given to 2 workers.