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Bismuth83
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When the focus group of 24 people are asked to consider two possible a 19 Nov 2024, 11:00
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Minimum number who recognized both: 4 Maximum possible number who recognized neither: 8
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68% (02:31) correct 32% (02:26) wrong based on 438 sessions

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When the focus group of 24 people are asked to consider two possible ads, they are also asked about the talent used in those ads. One ad features an on-camera celebrity endorser, but the celebrity may not be recognizable to all demographics. The other ad is an animated treatment with a voice provided by a comedian whose voice may or may not be recognized by viewers. Members of the focus group are asked if they recognize the celebrity endorser and if they recognize the voice in the animation. A total of 16 group members said they recognized the celebrity in the live ad, and 12 said they recognized the voice in the animation. Find the minimum number of respondents who recognized both and the maximum possible number who recognized neither.
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Minimum number who recognized both: 4 Maximum possible number who recognized neither: 8
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poojaarora1818
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When the focus group of 24 people are asked to consider two possible a 19 Nov 2024, 13:54
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Could you please provide the solution? I have tried but have been unable to find the correct solution. To my best understanding, it's a problem on overlapping sets where we need to minimize one value to maximize the other value.
Bismuth83
When the focus group of 24 people are asked to consider two possible ads, they are also asked about the talent used in those ads. One ad features an on-camera celebrity endorser, but the celebrity may not be recognizable to all demographics. The other ad is an animated treatment with a voice provided by a comedian whose voice may or may not be recognized by viewers. Members of the focus group are asked if they recognize the celebrity endorser and if they recognize the voice in the animation. A total of 16 group members said they recognized the celebrity in the live ad, and 12 said they recognized the voice in the animation. Find the minimum number of respondents who recognized both and the maximum possible number who recognized neither.
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SatvikVedala
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When the focus group of 24 people are asked to consider two possible a 19 Nov 2024, 18:41
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Could you please provide the solution? I have tried but have been unable to find the correct solution. To my best understanding, it's a problem on overlapping sets where we need to minimize one value to maximize the other value.
Bismuth83
When the focus group of 24 people are asked to consider two possible ads, they are also asked about the talent used in those ads. One ad features an on-camera celebrity endorser, but the celebrity may not be recognizable to all demographics. The other ad is an animated treatment with a voice provided by a comedian whose voice may or may not be recognized by viewers. Members of the focus group are asked if they recognize the celebrity endorser and if they recognize the voice in the animation. A total of 16 group members said they recognized the celebrity in the live ad, and 12 said they recognized the voice in the animation. Find the minimum number of respondents who recognized both and the maximum possible number who recognized neither.
Hi poojaarora1818

Following is my explanation for the problem

24 (Focus group) = 16 (Group1) + 12 (Group2) - x (Both) + N (neither)

24 = 28-x+N

x-N = 4

For N to be max x-4 needs to be max. Also we know that maximum value x can take is 12 (i.e. min(16,12))

12-4 = Nmax

Therefore, Nmax = 8

For x to be min N needs to be min. The minimum value N can take is 0

xmin = 4+Nmin => xmin = 4

Therefore xmin = 4 & Nmax = 8
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When the focus group of 24 people are asked to consider two possible a 19 Nov 2024, 18:59
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poojaarora1818
Could you please provide the solution? I have tried but have been unable to find the correct solution. To my best understanding, it's a problem on overlapping sets where we need to minimize one value to maximize the other value.
Bismuth83
When the focus group of 24 people are asked to consider two possible ads, they are also asked about the talent used in those ads. One ad features an on-camera celebrity endorser, but the celebrity may not be recognizable to all demographics. The other ad is an animated treatment with a voice provided by a comedian whose voice may or may not be recognized by viewers. Members of the focus group are asked if they recognize the celebrity endorser and if they recognize the voice in the animation. A total of 16 group members said they recognized the celebrity in the live ad, and 12 said they recognized the voice in the animation. Find the minimum number of respondents who recognized both and the maximum possible number who recognized neither.
Hi poojaarora1818

Following is my explanation for the problem

24 (Focus group) = 16 (Group1) + 12 (Group2) - x (Both) + N (neither)

24 = 28-x+N

x-N = 4

For N to be max x-4 needs to be max. Also we know that maximum value x can take is 12 (i.e. min(16,12))

12-4 = Nmax

Therefore, Nmax = 8

For x to be min N needs to be min. The minimum value N can take is 0

xmin = 4+Nmin => xmin = 4

Therefore xmin = 4 & Nmax = 8
Thank you so much for the detailed solution. Now, it brings more clarity to me and makes sense.
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When the focus group of 24 people are asked to consider two possible a 20 Nov 2024, 11:00
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1. We are given two characteristics that people either recognized or not. The question asks us to find the minimum number of respondents who recognized both and the maximum possible number who recognized neither.

2. Let's put the information in the text into a table:
Recognized celebrityDidn't recognize celebritySum
Recognized voicex?12
Didn't recognize voice?y?
Sum16?Total = 24

3. The question can be turned into finding the minimum of x and the maximum of y. Now, let's fill in some blanks:
Recognized celebrityDidn't recognize celebritySum
Recognized voicex12 - x12
Didn't recognize voice16 - xy?
Sum16?Total = 24

4. Since the total is 24, \(x + (16 - x) + (12 - x) + y = 24 \rightarrow y = x - 4\).

5. All the values must be positive, so, \(4 \leq x \leq 12\). The minimum of x is 4 and the maximum of y is when x is at its maximum, which is 12. That means y = 12 - 4 = 8.

6. Our answer will be: the minimum number of respondents who recognized both - 4; the maximum possible number who recognized neither - 8.
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