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12 Nov 2024, 04:07
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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
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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40%
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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.
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.
| True | False | |
| 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. |
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Official 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
12 Nov 2024, 04:07
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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"
Bunuel
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"
20 Nov 2024, 04:49
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