Comprehension

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
Graph
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?

Updated On: Nov 30, 2024
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Correct Answer: 50

Solution and Explanation

Day 1: Let the number of buyers on Day 1 be x.
The total rating on Day 1 = x.3 (since the average rating is 3)
Day 2: From the graph, we can see that: 10 buyers gave a rating of 1, 5 buyers gave a rating of 2, 15 buyers gave a rating of 3, 20 buyers gave a rating of 4, 25 buyers gave a rating of 5
Total buyers on Day 2 = 10+5+15+20+25 = 75
Total rating on Day 2 = (101) + (52) + (153) + (204) + (255) = 270
Cumulative Average:
Cumulative average after Day 2 = (Total rating on Day 1 + Total rating on Day 2) / (Total buyers on Day 1 + Total buyers on Day 2)
3.1=(3x+270)(x+75)3.1 = \frac {(3x + 270)}{(x + 75)}
Solving this equation, we get x=50x = 50.
Therefore, the number of buyers who gave ratings on Day 1 is 50.

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Question: 2

What is the daily average rating of Day 3?

Updated On: Nov 30, 2024
  • 3.6
  • 3.2
  • 3.5
  • 3.0
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The Correct Option is A

Solution and Explanation

Day 3 Ratings : Let the number of buyers giving ratings of 1 and 2 be x. Then, the number of buyers giving a rating of 3 is 2x. The remaining buyers (100 - 3x) must have given ratings of 4 and 5, with equal numbers of each (since 4 and 5 are the modes).
So, the distribution of ratings on Day 3 is: 1: x 2: x 3: 2x 4: (100-3x)/2 5: (100-3x)/2
Calculating the Average: The total rating on Day 3 is:
x 1 + x 2 + 2x 3 + (100-3x)/2 4 + (100-3x)/2 5
Simplifying, we get: 10x+45010x + 450.
The average rating on Day 3 is the total rating divided by the number of buyers: (10x+450)100\frac {(10x + 450)}{100}
To find the exact average, we need to find the value of x. However, we don’t have enough information to determine the exact value of x.
However, we can see that the average rating must be greater than 3.5 (since the majority of ratings are 4 and 5). The only option greater than 3.5 is 3.6.
Therefore, the daily average rating of Day 3 is approximately 3.6.

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Question: 3

What is the median of all ratings given on Day 3?

Updated On: Nov 30, 2024
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Correct Answer: 4

Solution and Explanation

Day 3 Ratings:
Let the number of buyers giving ratings of 1 and 2 be x. Then, the number of buyers giving a rating of 3 is 2x. The remaining buyers (100 - 3x) must have given ratings of 4 and 5, with equal numbers of each (since 4 and 5 are the modes).
So, the distribution of ratings on Day 3 is:
1: x 2: x 3: 2x 4: (100-3x)/2 5: (100-3x)/2
To find the median, we need to find the middle value when the ratings are arranged in ascending order.
Since there are 100 ratings, the median will be the average of the 50th and 51st ratings.
The first 3x ratings are 1, 2, and 3. The next (1003x)2\frac {(100-3x)}{2} ratings are 4.
So, the 50th and 51st ratings will be 4.

Therefore, the median of all ratings given on Day 3 is 4.

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Question: 4

Which of the following is true about the cumulative average ratings of Day 2 and Day 3?

Updated On: Nov 30, 2024
  • The cumulative average of Day 3 increased by more than 8% from Day 2
  • The cumulative average of Day 3 increased by a percentage between 5% and 8% from Day 2
  • The cumulative average of Day 3 decreased from Day 2.
  • The cumulative average of Day 3 increased by less than 5% from Day 2.
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The Correct Option is B

Solution and Explanation

Day 2: Total buyers: 75 Total rating: 270 Cumulative average: 3.1
Day 3: Total buyers: 100 Total rating: (From previous calculations, this is 450)
Cumulative average: (270+450)(75+100)=4\frac {(270 + 450)}{(75 + 100)}= 4
Percentage increase: (43.1)3.1×100=29.03%\frac {(4 - 3.1)}{3.1} \times100 = 29.03 \%

Therefore, the cumulative average of Day 3 increased by more than 8 percent from Day 2.

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