CAT 2024 Slot 2 DILR Question & Solution
Data Set
An online e-commerce firm receives daily integer product ratings from 1 through 5 given by buyers. The daily average is the average of the ratings given on that day. The cumulative average is the average of all ratings given on or before that day. The rating system began on Day 1, and the cumulative averages were 3 and 3.1 at the end of Day 1 and Day 2, respectively. The distribution of ratings on Day 2 is given in the figure below.
The following information is known about ratings on Day 3.
1. 100 buyers gave product ratings on Day 3.
2. The modes of the product ratings were 4 and 5.
3. The numbers of buyers giving each product rating are non-zero multiples of 10.
4. The same number of buyers gave product ratings of 1 and 2, and that number is half the number of buyers who gave a rating of 3.
Question 1
How many buyers gave ratings on Day 1?
Solution:
From the given chart, we can find the average rating on day 2:
$\frac{\left(5\times\ 1\right)+\left(10\times\ 2\right)+\left(5\times\ 3\right)+\left(20\times\ 4\right)+\left(10\times\ 5\right)}{5+10+5+20+10}=\frac{170}{50}=3.4$
We are given the cumulative average of day 1 and day 2 as 3.1, and the average at the end of day 1 is 3.
Let's take the number of ratings received on day 1 as $x$; using this overall average and the average on day 2, we get the equation:
$\frac{3x+\left(50\times\ 3.4\right)}{x+50}=3.1$
$3x+170=3.1x+155$
$15\ =\ 0.1x$
$x=150$
Therefore, the number of ratings received on day 1 is 150.
Question 2
What is the daily average rating of Day 3?
Solution:
We are given that on day 3, a total of 100 ratings came in
The modes were 4 and 5, meaning that an equal number of 4 and 5 ratings came in; let it be 10b (since we are given that all ratings were non-zero multiples of 10)
Let's take the number of 1 and 2 ratings as 10a each, giving the number of 3 ratings as 20a
Adding all of these up: 10a+10a+20a+10b+10b = 100
40a+20b = 100
2a+b = 5
The only integer combination for a and b, without them being zero, is a being 1 and b being 3 or a=2 and b=1
But taking b as 1 would make 3 as the mode rating, so we can not consider the latter case.
We are giving 10 - 1 ratings, 10 - 2 ratings, 20 - 3 ratings, 30 - 4 ratings, and 30 - 5 ratings.
The average would be $\frac{\left(10\times\ 1\right)+\left(10\times\ 2\right)+\left(20\times\ 3\right)+\left(30\times\ 4\right)+\left(30\times\ 5\right)}{100}=\frac{360}{100}=3.6$
Therefore, Option A is the correct answer.
Question 3
What is the median of all ratings given on Day 3?
Solution:
As deduced in the previous question, there were 10 1 ratings, 10 2 ratings, 20 3 ratings, 30 4 ratings, and 30 5 ratings.
The median would be the 50th and 51st ratings' average when arranged in ascending/descending order.
Both of which would be 4
Therefore, 4 is the correct answer
Question 4
Which of the following is true about the cumulative average ratings of Day 2 and Day 3?
Solution:
The cumulative average rating at the end of day 3 would be $\frac{\left(3.1\times\ 200\right)+\left(3.6\times\ 100\right)}{200+100}$
[the cumulative average rating on day 2= 3.1
Number of ratings received by day 2= 200]
$\frac{360+620}{300}=\frac{980}{300}=3.266$
The increase in the cumulative average from day 2 to day 3 can be calculated as $\frac{3.266-3.1}{3.1}\times\ 100\approx\ 5.34$
Which aligns with the statement given in option C.
Therefore, Option C is the correct answer.