Question

How many buyers gave ratings on Day 1?

• A. 3.6
• B. 3.2
• C. 3.5
• D. 3.0
• A. The cumulative average of Day 3 increased by more than 8% from Day 2
• B. The cumulative average of Day 3 increased by a percentage between 5% and 8% from Day 2
• C. The cumulative average of Day 3 decreased from Day 2.
• D. The cumulative average of Day 3 increased by less than 5% from Day 2.

Answer 1

Correct Answer: 50

Day 1: Let the number of buyers on Day 1 be \( x \). 
The total rating on Day 1 = \( 3x \) (since the average rating is 3).

Day 2: From the graph, we can see:
- 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 = \( (10 \cdot 1) + (5 \cdot 2) + (15 \cdot 3) + (20 \cdot 4) + (25 \cdot 5) = 270 \)

Cumulative Average:
Cumulative average after Day 2 = \[ \frac{3x + 270}{x + 75} = 3.1 \]

Solving this equation:

\[ 3.1(x + 75) = 3x + 270 \\ 3.1x + 232.5 = 3x + 270 \\ 0.1x = 37.5 \Rightarrow x = 375 \]

Correct value: \( x = 375 \)
Therefore, the number of buyers who gave ratings on Day 1 is 375.

Answer 2

To find the daily average rating for Day 3, we follow these steps:

1. Let the number of buyers giving a rating of 1 or 2 be x. Then, the number of buyers giving a rating of 3 is 2x.
2. The numbers of buyers giving ratings of 4 and 5, according to the given information, are non-zero multiples of 10. Let the number of buyers giving these ratings be y and z, respectively.
3. Since the modes of the ratings are 4 and 5, both y and z must be greater than or equal to 2x.
4. The total number of buyers on Day 3 is 100. Hence, we have the equation:
   x + x + 2x + y + z = 100
   which simplifies to:
   4x + y + z = 100
5. Additionally, since y and z are equal (as they are the modes and we assume the most straightforward case), we can set y = z.
6. Further simplifying using y = z, we substitute into 4x + 2y = 100 to get:
   2x + y = 50
7. From the condition of non-zero multiples of 10 and that y and z are the largest, we assume y = z = 40 and solve:
   2x + 40 = 50
   This gives us 2x = 10, so x = 5.
8. Now we distribute the ratings as follows: 5 buyers rated 1, 5 buyers rated 2, 10 buyers rated 3, and 40 buyers each for ratings 4 and 5.
9. The formula for the daily average rating is given by:
   (1*5 + 2*5 + 3*10 + 4*40 + 5*40) / 100
10. Calculating each part, we get:
    5 + 10 + 30 + 160 + 200 = 405
11. Thus, the daily average rating is:
    405 / 100 = 4.05.

Answer 3

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: \(\frac{100 - 3x}{2}\), 5: \(\frac{100 - 3x}{2}\)

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 50^th and 51^st ratings.
The first \(3x\) ratings are 1, 2, and 3. The next \(\frac{100 - 3x}{2}\) ratings are 4.
So, the 50^th and 51^st ratings will be 4.

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

Answer 4

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: \(\frac {(270 + 450)}{(75 + 100)}= 4\)
Percentage increase: \(\frac {(4 - 3.1)}{3.1} \times100 = 29.03 \%\)

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