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14 Feb 2024, 07:52
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The value of \(x\), representing the range of ratings Don Pizza received, was 4.: Fals
Pizza King received exactly one rating of 5.: True
There are exactly two distinct possible distributions of the five ratings that Don Pizza could have received.: Fals
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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45%
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In today's digital food market, a simple 5-point rating system carries significant weight. These ratings often serve as key filters, helping to narrow choices and provide an instant snapshot of a restaurant's performance.
On a particular day, two restaurants received exactly five ratings each, as shown in the table. The ratings are integers, ranging from 1 to 5 inclusive, and are analyzed using the following metrics:
• The Range, which is the difference between the highest and lowest values;
• The Mean, calculated as the average of the five ratings;
• The Median, which is the middle value when the ratings are arranged in ascending or descending order;
• The Mode, the value that appears most frequently.
For each of the following statements, select True if it can be verified as correct based on the information provided. Otherwise, select False. Male only two selections, one in each column.
On a particular day, two restaurants received exactly five ratings each, as shown in the table. The ratings are integers, ranging from 1 to 5 inclusive, and are analyzed using the following metrics:
• The Range, which is the difference between the highest and lowest values;
• The Mean, calculated as the average of the five ratings;
• The Median, which is the middle value when the ratings are arranged in ascending or descending order;
• The Mode, the value that appears most frequently.
| Statistical Data | Don Pizza | Pizza King |
|---|---|---|
| Mean | 3.2 | 3.4 |
| Median | 2 | 3 |
| Mode | 2 | 3 |
| Range | x | 3 |
For each of the following statements, select True if it can be verified as correct based on the information provided. Otherwise, select False. Male only two selections, one in each column.
| True | Fals | |
| The value of \(x\), representing the range of ratings Don Pizza received, was 4. | ||
| Pizza King received exactly one rating of 5. | ||
| There are exactly two distinct possible distributions of the five ratings that Don Pizza could have received. |
ShowHide Answer
Official Answer
The value of \(x\), representing the range of ratings Don Pizza received, was 4.: Fals
Pizza King received exactly one rating of 5.: True
There are exactly two distinct possible distributions of the five ratings that Don Pizza could have received.: Fals
20 Nov 2024, 04:53
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Bunuel
Official Solution:
Statement 1:
The mean of Don Pizza’s rating being 3.2 implies that the sum of the five ratings was 3.2 * 5 = 16.
The median of the ratings being 2 means that 2 is the middle rating: \(\{a, b, 2, c, d\}\).
The mode of the ratings being 2 indicates that at least one other rating was also 2.
If the range of the ratings was 4, then the lowest rating must be 1, and the highest must be 5: \(\{1, b, 2, c, 5\}\).
Since at least one more rating must be 2 (from the mode), the ratings would include 1, 2, 2, and 5, summing to 10. To reach a total of 16, the fifth rating would need to be 6, which is not possible because the highest rating is 5. Therefore, this statement is False.
Statement 2:
The mean of Pizza King’s rating being 3.4 implies that the sum of the five ratings was 3.4 * 5 = 17.
The median of the ratings being 3 means that 3 is the middle rating: \(\{a, b, 3, c, d\}\).
The mode of the ratings being 3 indicates that at least one other rating was also 3.
If the range of the ratings was 3, then the lowest rating must be \(a\), and the highest must be \(d = a + 3\): \(\{a, b, 3, c, a+3\}\).
Let's check if Pizza King could have received two ratings of 5. That would mean that \(a=2\), \(c=5\), and \(a+3=5\), and since the mode is 3, \(b=3\): \(\{2, 3, 3, 5, 5\}\). The sum of these ratings is 18, not 17.
Let's check if Pizza King could have received no rating of 5 and still got the sum of the ratings equal to 17. The maximum sum would be if \(d = 4\), and from the range being 3, \(a\) would become 1. The mode of 3 would make \(b\) equal 3, and since we are maximizing the sum, \(c\) would become 4 too: \(\{1, 3, 3, 4, 4\}\). The sum of these ratings is 15, not 17.
Therefore, Pizza King must have received exactly one rating of 5, with the distribution of scores as \(\{2, 3, 3, 4, 5\}\).
Statement 3:
From the analysis of the first statement, we determined that the range must have been 3 or less. Let's calculate the maximum possible sum of the ratings within this range, given the median of 2 and the mode of 2 for \(\{a, b, 2, c, d\}\). This results in the following distribution: \(\{2, 2, 2, 5, 5\}\). The sum of this distribution is 16, which matches the actual sum. Any change in this distribution would decrease the sum and fail to satisfy the conditions.
Since only one valid distribution, \(\{2, 2, 2, 5, 5\}\), satisfies all the given conditions, there are not two distinct possible distributions. Therefore, this statement is False.
Correct answer:
The value of \(x\), representing the range of ratings Don Pizza received, was 4. "False"
Pizza King received exactly one rating of 5. "True"
There are exactly two distinct possible distributions of the five ratings that Don Pizza could have received. "False"
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General Discussion
03 Jan 2025, 16:31
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1. Let's make a table out of the ratings and order them from left to right as increasing.
2. Rating 3 for Don Pizza and Pizza King must be the median and is 2 and 3, respectively.
3. Pizza King as an average of 3.4 or a sum (of ratings) 3.4 \(*\) 5 = 17. If there was no rating of 5, then the maximum sum that could be achieved is by having [3,3,3,4,4], so 17. However the range is 2 when it should be 3. Changing it would only make the sum lower. So, there must be at least one 5. That means all ratings must be from 2 to 5 and rating 1 is 2.
4. Rating 2 or/and rating 4 must be 3 for the mode requirement to work. If rating 4 was 3, then the maximum sum would be 16 from [2,3,3,3,5], which doesn't work. So, rating 2 must be 3 and rating 4 is 17 - 5 - 3 - 3 - 2 = 4.
5. Now let's look at Don Pizza. The average is 3.2, so 3.2 \(*\) 5 = 16 is the sum of all ratings. The only restriction left is the mode, which is 2. That means rating 2 or/and rating 4 is 2. If rating 4 is 2, then the maximum sum is 13 from [2,2,2,2,5], which doesn't work. So, rating 2 is 2.
6. If rating 1 is 1, then the maximum sum is 15 from [1,2,2,5,5], which doesn't work. So, rating 1 is 2.
7. The minimum sum is 16 from [2,2,2,4,4]. With only ratings 4 and 5 being able to change, the sum will only increase. So, both of them must be equal to 4.
8. Now let's check whether the following statements are true or false.
9. The value of x, representing the range of ratings Don Pizza received, was 4. The range for Don Pizza is 4 - 2 = 2. So, this statement is false.
10. Pizza King received exactly one rating of 5. Only rating 5 has a 5 for Pizza King. So, this statement is true.
11. There are exactly two distinct possible distributions of the five ratings that Don Pizza could have received. We arrived at only one possible distribution for Don Pizza. So, this statement is false.
| Rating 1 | Rating 2 | Rating 3 | Rating 4 | Rating 5 | |
| Don Pizza | ? | ? | ? | ? | ? |
| Pizza King | ? | ? | ? | ? | ? |
2. Rating 3 for Don Pizza and Pizza King must be the median and is 2 and 3, respectively.
| Rating 1 | Rating 2 | Rating 3 | Rating 4 | Rating 5 | |
| Don Pizza | ? | ? | 2 | ? | ? |
| Pizza King | ? | ? | 3 | ? | ? |
3. Pizza King as an average of 3.4 or a sum (of ratings) 3.4 \(*\) 5 = 17. If there was no rating of 5, then the maximum sum that could be achieved is by having [3,3,3,4,4], so 17. However the range is 2 when it should be 3. Changing it would only make the sum lower. So, there must be at least one 5. That means all ratings must be from 2 to 5 and rating 1 is 2.
| Rating 1 | Rating 2 | Rating 3 | Rating 4 | Rating 5 | |
| Don Pizza | ? | ? | 2 | ? | ? |
| Pizza King | 2 | ? | 3 | ? | 5 |
4. Rating 2 or/and rating 4 must be 3 for the mode requirement to work. If rating 4 was 3, then the maximum sum would be 16 from [2,3,3,3,5], which doesn't work. So, rating 2 must be 3 and rating 4 is 17 - 5 - 3 - 3 - 2 = 4.
| Rating 1 | Rating 2 | Rating 3 | Rating 4 | Rating 5 | |
| Don Pizza | ? | ? | 2 | ? | ? |
| Pizza King | 2 | 3 | 3 | 4 | 5 |
5. Now let's look at Don Pizza. The average is 3.2, so 3.2 \(*\) 5 = 16 is the sum of all ratings. The only restriction left is the mode, which is 2. That means rating 2 or/and rating 4 is 2. If rating 4 is 2, then the maximum sum is 13 from [2,2,2,2,5], which doesn't work. So, rating 2 is 2.
| Rating 1 | Rating 2 | Rating 3 | Rating 4 | Rating 5 | |
| Don Pizza | ? | 2 | 2 | ? | ? |
| Pizza King | 2 | 3 | 3 | 4 | 5 |
6. If rating 1 is 1, then the maximum sum is 15 from [1,2,2,5,5], which doesn't work. So, rating 1 is 2.
| Rating 1 | Rating 2 | Rating 3 | Rating 4 | Rating 5 | |
| Don Pizza | 2 | 2 | 2 | ? | ? |
| Pizza King | 2 | 3 | 3 | 4 | 5 |
7. The minimum sum is 16 from [2,2,2,4,4]. With only ratings 4 and 5 being able to change, the sum will only increase. So, both of them must be equal to 4.
| Rating 1 | Rating 2 | Rating 3 | Rating 4 | Rating 5 | |
| Don Pizza | 2 | 2 | 2 | 4 | 4 |
| Pizza King | 2 | 3 | 3 | 4 | 5 |
8. Now let's check whether the following statements are true or false.
9. The value of x, representing the range of ratings Don Pizza received, was 4. The range for Don Pizza is 4 - 2 = 2. So, this statement is false.
10. Pizza King received exactly one rating of 5. Only rating 5 has a 5 for Pizza King. So, this statement is true.
11. There are exactly two distinct possible distributions of the five ratings that Don Pizza could have received. We arrived at only one possible distribution for Don Pizza. So, this statement is false.
