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Zorluk: Çok zorOverlapping Sets, Statistics, and Data Distributions

In a cohort of NN financial analysts, every analyst tracks at least one of two asset classes: Equities or Bonds. The mean number of years of experience for analysts who track Equities is 12 years, and the mean number of years of experience for analysts who track Bonds is 18 years. Is the mean number of years of experience for all NN analysts in the cohort strictly greater than 15 years?

(1) The mean number of years of experience for analysts who track both Equities and Bonds is 15 years.
(2) The number of analysts who track Bonds only is strictly greater than the number of analysts who track Equities only.

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Cevap
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Both statements together are sufficient to answer the question, but neither statement alone is sufficient.
Rephrasing the question stem shows that the overall mean is greater than 15 if and only if 3(y−x)+(15−mˉ)z>03(y - x) + (15 - \bar{m})z > 0, where xx is analysts tracking Equities only, yy is analysts tracking Bonds only, zz is analysts tracking both, and mˉ\bar{m} is the mean experience of the overlap group. Statement (1) gives mˉ=15\bar{m} = 15, simplifying the target to y>xy > x, which is insufficient alone. Statement (2) gives y>xy > x, which is insufficient alone because mˉ\bar{m} is unknown. Combined, Statement (1) simplifies the target to y>xy > x and Statement (2) confirms y>xy > x, yielding a definitive 'Yes'.

Adım Adım Çözüm

1
Define variables and set up the expression for the total sum of experience and overall mean.
Let xx be the number of analysts tracking Equities only, yy be the number of analysts tracking Bonds only, and zz be the number of analysts tracking both. Total analysts N=x+y+zN = x + y + z.
Sum of experience for Equities group: SE=12(x+z)S_E = 12(x+z).
Sum of experience for Bonds group: SB=18(y+z)S_B = 18(y+z).
Let mˉ\bar{m} be the mean experience of the zz analysts tracking both, so their total experience is zmˉz\bar{m}.
Total experience Stotal=SE+SB−zmˉ=12x+18y+(30−mˉ)zS_{total} = S_E + S_B - z\bar{m} = 12x + 18y + (30 - \bar{m})z.
Overlapping sets counting principle requires subtracting the overlap sum so double-counted individuals are accounted for correctly.
2
Rephrase the target question stem inequality.
The overall mean is Mˉ=StotalN=12x+18y+(30−mˉ)zx+y+z\bar{M} = \frac{S_{total}}{N} = \frac{12x + 18y + (30 - \bar{m})z}{x + y + z}.
The question asks if Mˉ>15\bar{M} > 15:
12x+18y+(30−mˉ)zx+y+z>15\frac{12x + 18y + (30 - \bar{m})z}{x + y + z} > 15
12x+18y+(30−mˉ)z>15x+15y+15z12x + 18y + (30 - \bar{m})z > 15x + 15y + 15z
3y−3x+(15−mˉ)z>0  ⟺  3(y−x)+(15−mˉ)z>03y - 3x + (15 - \bar{m})z > 0 \iff 3(y - x) + (15 - \bar{m})z > 0
Simplifying the target question algebraically reveals the exact algebraic condition required to determine sufficiency.
3
Evaluate Statement (1) alone.
Statement (1) states mˉ=15\bar{m} = 15.
Substituting mˉ=15\bar{m} = 15 into the rephrased inequality yields 3(y−x)+(15−15)z>0  ⟺  3(y−x)>0  ⟺  y>x3(y - x) + (15 - 15)z > 0 \iff 3(y - x) > 0 \iff y > x.
Since we do not know whether y>xy > x or y≤xy \leq x, Statement (1) alone is NOT sufficient.
Knowing the mean of the overlap eliminates the zz term but leaves the relationship between xx and yy unknown.
4
Evaluate Statement (2) alone.
Statement (2) states y>xy > x.
Without knowing the value of mˉ\bar{m} (the mean of the overlapping group), if mˉ\bar{m} is very large (e.g., mˉ=50\bar{m} = 50), then (15−mˉ)z(15 - \bar{m})z could be negative enough to make 3(y−x)+(15−mˉ)z<03(y - x) + (15 - \bar{m})z < 0. Thus, Statement (2) alone is NOT sufficient.
Without information about the overlapping group's mean, the sign of 3(y−x)+(15−mˉ)z3(y - x) + (15 - \bar{m})z cannot be determined.
5
Evaluate Statements (1) and (2) together.
From Statement (1), the condition simplifies to y>xy > x.
From Statement (2), we are explicitly given that y>xy > x.
Therefore, together the statements definitively prove that the overall mean is strictly greater than 15 years (Yes). Both statements together are SUFFICIENT.
Combining both statements satisfies the rephrased target condition completely.

Anahtar Kavram

Overlapping Sets and Weighted Averages in Data Sufficiency
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Overlapping Sets, Statistics, and Data Distributions

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