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­In this digital world of the food market, a simple 5-point rating sys

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chetan2u
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­In this digital world of the food market, a simple 5-point rating sys [#permalink] New post   14 Feb 2024, 07:52
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Data Insights (DI) Butler 2024 [ Date: 14-Feb-2023] 


­In this digital world of the food market, a simple 5-point rating system holds immense weight. Ratings often act as primary filters, helping to narrow down choices and offering an immediate snapshot of a restaurant's performance.

On a particular day, two restaurants got exactly five ratings as given in the table. The ratings were given out of digits 1 to 5, where the Range is the difference between the largest and smallest value. The Mean is the average of the five ratings, while the Median is the center value when the values are placed in ascending/descending order, and the Mode is the value that appears the maximum number of times.

Statistical DataDumb PizzaPizza King
Mean3.23.4
Median23
Mode23
Rangex3

(Sort ↕ the table by clicking on the headers)

For each of the following statements, select Yes if that statement accurately reflects the information provided. Otherwise, select No.­­­­­­­
Yes No
The value of x, that is the range of ratings that Dumb Pizza got was 4
The maximum rating that Pizza King could have got was 5
There are exactly two different possible combinations of the five ratings that Dumb Pizza could have got.
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The value of x, that is the range of ratings that Dumb Pizza got was 4: No The maximum rating that Pizza King could have got was 5: Yes There are exactly two different possible combinations of the five ratings that Dumb Pizza could have got.: No
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Re: ­In this digital world of the food market, a simple 5-point rating sys [#permalink] New post   15 Feb 2024, 05:04
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chetan2u wrote:
Data Insights (DI) Butler 2024 [ Date: 14-Feb-2023] 


­In this digital world of the food market, a simple 5-point rating system holds immense weight. Ratings often act as primary filters, helping to narrow down choices and offering an immediate snapshot of a restaurant's performance.

On a particular day, two restaurants got exactly five ratings as given in the table. The ratings were given out of digits 1 to 5, where the Range is the difference between the largest and smallest value. The Mean is the average of the five ratings, while the Median is the center value when the values are placed in ascending/descending order, and the Mode is the value that appears the maximum number of times.

Statistical DataDumb PizzaPizza King
Mean3.23.4
Median23
Mode23
Rangex3

(Sort ↕ the table by clicking on the headers)

For each of the following statements, select Yes if that statement accurately reflects the information provided. Otherwise, select No.­­­­

­chetan2u Can you please share the Official Explanation especially for the 3rd prompt
As per my understanding, for Dumb Pizza : Total = 16 (3.2*5) is achievable by 2, 2, 2, 5, 5 only. Am I missing anyother combination ? or is it entirely a wrong set ?
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­In this digital world of the food market, a simple 5-point rating sys [#permalink] New post   15 Feb 2024, 05:37
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Dumb Pizza
Mean-3.2, Median-2, Mode-2
Therefore, sum of the ratings = 3.2*5 = 16.
However, there are at least two 2s in it, thus sum of remaining three = 16-4 = 12.
Now, 12 can be made from (3,4,5), (2,5,5) and (4,4,4).
(3,4,5) will change the median to 3 => 2,2,3,4,5
(4,4,4) will change the median and mode to 4 => 2,2,4,4,4
(2,5,5) is the only possibility => 2,2,2,5,5
Thus, range is 5-2 =3, and the possibility is only one.
1st case- No and 3rd case - No

Pizza King
Mean-3.4, Median-3, Mode-3
Therefore, sum of the ratings = 3.4*5 = 17.
However, there are at least two 3s in it, thus sum of remaining three = 17-6 = 11.
Now, 11 can be made from (1,5,5), (2,4,5) and (3,4,4).
(1,5,5) will change the mode to (3,5) => 1,3,3,5,5...Not valid as mode changes
(2,4,5) will be a correct possibility => 2,3,3,4,5
(3,4,4) is another possibility => 3,3,3,4,4......Not valid a Range changes
So, one possibility and maximum rating can be 5.
Thus, case 2 is yes.

Aryaa03, you are absolutely correct in your observation.­
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Re: ­In this digital world of the food market, a simple 5-point rating sys [#permalink] New post   15 Feb 2024, 06:37
chetan2u wrote:
Dumb Pizza
Mean-3.2, Median-2, Mode-2
Therefore, sum of the ratings = 3.2*5 = 16.
However, there are at least two 2s in it, thus sum of remaining three = 16-4 = 12.
Now, 12 can be made from (3,4,5), (2,5,5) and (4,4,4).
(3,4,5) will change the median to 3 => 2,2,3,4,5
(4,4,4) will change the median and mode to 4 => 2,2,4,4,4
(2,5,5) is the only possibility => 2,2,2,5,5
Thus, range is 5-2 =3, and the possibility is only one.
1st case- No and 3rd case - No

Pizza King
Mean-3.4, Median-3, Mode-3
Therefore, sum of the ratings = 3.4*5 = 17.
However, there are at least two 3s in it, thus sum of remaining three = 17-6 = 11.
Now, 11 can be made from (1,5,5), (2,4,5) and (3,4,4).
(1,5,5) will change the mode to (3,5) => 1,3,3,5,5
(2,4,5) will be a correct possibility => 2,3,3,4,5
(3,4,4) is the another possibility => 3,3,3,4,4
So, two possibilities and maximum rating can be 5.
Thus, case 2 is yes.

Aryaa03, you are absolutely correct in your observation.­


Thank you for the prompt response chetan2u. Great explanation as always.

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Re: ­In this digital world of the food market, a simple 5-point rating sys [#permalink] New post   06 Mar 2024, 17:54
Hi chetan2u, what if there are at least three or four 2s in the first case? Since the mode is 2:)

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Re: ­In this digital world of the food market, a simple 5-point rating sys [#permalink] New post   07 Mar 2024, 00:27
Dumb Pizza:Let's build cases with sum of rating=16, and mid value=2, and atleast one more rating of 2

Case 1: 2,2,2,4,6
Case 2: 1,2,2,4,7
Case 3: 1,2,2,3,8

Pizza King:
Case: 2,3,3,4,5

Ans: No, Yes, No
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­In this digital world of the food market, a simple 5-point rating sys [#permalink] New post   Updated on: 24 Apr 2024, 19:54
Number the ratings from lowest to highest
Dumb: \(d_1\) to \(d_5\)
King: \(k_1\) to \(k_5\)

­1. The value of x, that is the range of ratings that Dumb Pizza got was 4 => No

Total = 3.2 * 5 = 16

Median = 2 => \(m_3 = 2\)

Mode = 2 => at least \(m_2\) = 2 (\(m_2\) cannot be 1 since in that case, we cannot reach the total of 16)

\(m_1\) + \(m_4\) + \(m_5\) = 16 - 2 * 2 = 12

2 5 5 is the only possible combination => Range = 3


2. The maximum rating that Pizza King could have got was 5 => Yes

Total = 3.4 * 5 = 17

Median = 3 => \(k_3 = 3\)

Mode = 3 => at least 2 ratings are 3

Possible combinations:
  • 3 3 3 3 5
  • ...

3. There are exactly two different possible combinations of the five ratings that Dumb Pizza could have got. => No

As worked out in (1), there is only 1 possiblee combination­

Originally posted by Gemmie on 29 Mar 2024, 14:41.
Last edited by Gemmie on 24 Apr 2024, 19:54, edited 1 time in total.
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Re: ­In this digital world of the food market, a simple 5-point rating sys [#permalink] New post   24 Apr 2024, 10:53
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chetan2u wrote:
Pizza King
Mean-3.4, Median-3, Mode-3
Therefore, sum of the ratings = 3.4*5 = 17.
However, there are at least two 3s in it, thus sum of remaining three = 17-6 = 11.
Now, 11 can be made from (1,5,5), (2,4,5) and (3,4,4).
(1,5,5) will change the mode to (3,5) => 1,3,3,5,5
(2,4,5) will be a correct possibility => 2,3,3,4,5
(3,4,4) is the another possibility => 3,3,3,4,4
So, two possibilities and maximum rating can be 5.
Thus, case 2 is yes.



 

chetan2u ­In this case for Pizza King, (3,4,4) is the another possibility => 3,3,3,4,4  But the range for Pizza King is given as 3, So in case of Pizza King I think only 1 possibility is there?­
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Re: ­In this digital world of the food market, a simple 5-point rating sys [#permalink] New post   26 Apr 2024, 06:28
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Anugmat740 wrote:
chetan2u wrote:
Pizza King
Mean-3.4, Median-3, Mode-3
Therefore, sum of the ratings = 3.4*5 = 17.
However, there are at least two 3s in it, thus sum of remaining three = 17-6 = 11.
Now, 11 can be made from (1,5,5), (2,4,5) and (3,4,4).
(1,5,5) will change the mode to (3,5) => 1,3,3,5,5
(2,4,5) will be a correct possibility => 2,3,3,4,5
(3,4,4) is the another possibility => 3,3,3,4,4
So, two possibilities and maximum rating can be 5.
Thus, case 2 is yes.
 

chetan2u ­In this case for Pizza King, (3,4,4) is the another possibility => 3,3,3,4,4  But the range for Pizza King is given as 3, So in case of Pizza King I think only 1 possibility is there?­

Yes, you are correct. May be a typo as I don't remember why did I write two possibilities although it doesnt change the answer.­
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Re: ­In this digital world of the food market, a simple 5-point rating sys [#permalink]
26 Apr 2024, 06:28
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