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

Day 2: The graph shows:

• 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:
The cumulative average after Day 2 is \[ \frac{3x + 270}{x + 75} = 3.1 \]

Solving the equation:

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

Correct value: \( x = 375 \)
Thus, 375 buyers gave ratings on Day 1.

Answer 2

To find the daily average rating for Day 3, we proceed as follows:

1. Let x be the number of buyers who gave a rating of 1 or 2. The number of buyers who gave a rating of 3 is then 2x.
2. The numbers of buyers who gave ratings of 4 and 5 are non-zero multiples of 10. Let these numbers be y and z, respectively.
3. Since ratings 4 and 5 are the modes, both y and z must be at least 2x.
4. The total number of buyers on Day 3 is 100. This gives us the equation:
   x + x + 2x + y + z = 100
   Simplifying, we get:
   4x + y + z = 100
5. Assuming the most straightforward case where the modes are equal, we set y = z.
6. Substituting y = z into the equation from step 4, we get 4x + 2y = 100. Dividing by 2, we have:
   2x + y = 50
7. Given that y and z are the largest counts and multiples of 10, we assume y = z = 40 and solve for x:
   2x + 40 = 50
   This yields 2x = 10, so x = 5.
8. The distribution of ratings is: 5 buyers gave a rating of 1, 5 buyers gave a rating of 2, 10 buyers gave a rating of 3, and 40 buyers each gave ratings of 4 and 5.
9. The formula for the daily average rating is:
   (1*5 + 2*5 + 3*10 + 4*40 + 5*40) / 100
10. Calculating the sum of the weighted ratings:
    5 + 10 + 30 + 160 + 200 = 405
11. Therefore, the daily average rating is:
    405 / 100 = 4.05.

Answer 3

Day 3 Ratings:Nbsp;
Let \(x\) be the number of buyers who gave ratings of 1 and 2. Then, \(2x\) buyers gave a rating of 3. The remaining buyers, \((100 - 3x)\), gave ratings of 4 and 5. Since 4 and 5 are the modes (most frequent ratings), there must be an equal number of buyers for each.
Thus, the rating distribution on Day 3 is as follows:
1: \(x\), 2: \(x\), 3: \(2x\), 4: \(\frac{100 - 3x}{2}\), 5: \(\frac{100 - 3x}{2}\)

To determine the median, we find the middle value in the ordered list of ratings. With 100 ratings, the median is the average of the 50th and 51st ratings.
The first \(3x\) ratings consist of 1s, 2s, and 3s. The ratings following these are 4s, and there are \(\frac{100 - 3x}{2}\) such ratings.
Therefore, both the 50th and 51st ratings will be 4.

Consequently, the median of all ratings 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: 450.
Cumulative average: \(\frac {(270 + 450)}{(75 + 100)}= 4\)
Percentage increase:Nbsp;\(\frac {(4 - 3.1)}{3.1} \times100 = 29.03 \%\)

The cumulative average on Day 3 increased by over 8 percent compared to Day 2.