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19 Nov 2024, 11:00
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Minimum number who recognized both: 4
Maximum possible number who recognized neither: 8
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
65% (02:29) correct
35%
(02:37)
wrong
based on 1103
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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.
| Minimum number who recognized both | Maximum possible number who recognized neither | |
| 0 | ||
| 2 | ||
| 4 | ||
| 6 | ||
| 8 | ||
| 10 |
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Official Answer
Minimum number who recognized both: 4
Maximum possible number who recognized neither: 8
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:
3. The question can be turned into finding the minimum of x and the maximum of y. Now, let's fill in some blanks:
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.
2. Let's put the information in the text into a table:
| Recognized celebrity | Didn't recognize celebrity | Sum | |
| Recognized voice | x | ? | 12 |
| Didn't recognize voice | ? | y | ? |
| Sum | 16 | ? | 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 celebrity | Didn't recognize celebrity | Sum | |
| Recognized voice | x | 12 - x | 12 |
| Didn't recognize voice | 16 - x | y | ? |
| Sum | 16 | ? | 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.
19 Nov 2024, 18:41
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poojaarora1818
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
