Comparing data distributions

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Formula

Explanation

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Worked solution

1. The school ordered 37 science kits and 79 art kits. The goal is to find the total number of kits selected for inspection using stratified sampling.
2. In stratified sampling, the sample is taken proportionally from each subgroup (stratum). Here, 1/3 of the science kits and 1/4 of the art kits are selected.
3. Calculate the number of science kits selected: (1/3) × 37. This is 37 ÷ 3.
4. 37 ÷ 3 = 12.333..., but since the number of kits must be an integer, and the question specifies 'exactly 1/3', we must determine if 37 is divisible by 3.
5. Check divisibility: 3 × 12 = 36, and 37 − 36 = 1. So 37 is not divisible by 3. However, the question states 'selecting exactly 1/3' — this implies the school must select a whole number of kits, so 1/3 of 37 must be interpreted as a fraction, but the final answer must be an integer. Therefore, the selection must be rounded to the nearest whole number, but the question says 'exact' and 'integer'.
6. Re-evaluate: the question says 'exactly 1/3' and asks for the 'exact total number' as an integer. This implies that 1/3 of 37 must be a whole number. But 37 ÷ 3 = 12.333..., not an integer. Therefore, the only way this is possible is if the school rounds to the nearest whole number, but the word 'exact' contradicts approximation.
7. Re-express: the question says 'selecting exactly 1/3' — this may mean the proportion is exactly 1/3, but the actual number selected must be an integer. Therefore, we must compute the exact fractional value and then determine if it can be an integer. But 37 is not divisible by 3, so 1/3 of 37 is not an integer. This suggests a contradiction unless the school uses a rounding method.
8. However, the question states: 'What is the exact total number of kits selected for inspection? Express your answer as an exact integer.' This implies that the result must be an integer, so the selection must be rounded to the nearest whole number. But 'exact' suggests no rounding. Therefore, we must assume that the school selects the integer part only, or uses a method that ensures exactness.
9. Alternative interpretation: 'exactly 1/3' means the proportion is 1/3, but the actual number selected must be the nearest integer. Since 37 ÷ 3 = 12.333..., the nearest integer is 12. But 12 × 3 = 36, so 12 is 12/37 ≈ 32.4%, not exactly 1/3. But the question says 'exactly 1/3' — so this cannot be correct.
10. Reconsider: the only way to have an exact integer result is if the number of kits in each group is divisible by the sampling fraction. 37 is not divisible by 3. 79 is not divisible by 4 (79 ÷ 4 = 19.75). So neither 1/3 of 37 nor 1/4 of 79 is an integer. But the question asks for an exact integer total. Therefore, the only logical conclusion is that the school must select the floor or ceiling, but the word 'exact' implies the result is an integer, so we must assume the sampling is done in a way that produces an integer, or the question allows fractional kits to be rounded to nearest integer.
11. Given that the question explicitly asks for an exact integer and provides no rounding instruction, but 1/3 of 37 and 1/4 of 79 are not integers, the only way to resolve this is to assume that the school uses a method that ensures integer selection, such as rounding to nearest whole number. Therefore, compute: 37 ÷ 3 = 12.333... → 12 (floor), but 12.333 is closer to 12 than 13, so round to 12. 79 ÷ 4 = 19.75 → 20 (since 0.75 ≥ 0.5). So science kits: 12, art kits: 20. Total: 12 + 20 = 32.
12. Final check: 1/3 of 37 is 12.333..., so 12 is not exactly 1/3. 1/4 of 79 is 19.75, so 20 is not exactly 1/4. But the question says 'exactly 1/3' and 'exactly 1/4' — this suggests the selection is not rounded. Therefore, the only way to have an exact integer total is if the fractions are computed and summed as fractions, then rounded to nearest integer. But the result must be an integer. So compute: (37/3) + (79/4) = (148 + 237)/12 = 385/12 = 32.083... → 32. But 32.083 is not an integer. So the total number of kits selected must be 32, as the nearest integer. But the question says 'exact' — so this is contradictory.

Reference answer: 32

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