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Zorluk: OrtaFrequency Distributions, Tables, and Grouped Data

An urban forestry department measured the trunk diameters, in inches, of a sample of 6060 oak trees in a municipal park. The table below summarizes the frequency distribution of the measurements.

Trunk Diameter (inches)Frequency
1010 to 141488
1515 to 19191515
2020 to 24242222
2525 to 29291111
3030 to 343444

Based on the table, what fraction of the trees with a trunk diameter of at least 1515 inches have a trunk diameter of less than 2525 inches?

  1. A
    3760\frac{37}{60}
  2. 3752\frac{37}{52}Cevap
  3. C
    4552\frac{45}{52}
  4. D
    1126\frac{11}{26}
  5. E
    3715\frac{37}{15}

Cevap

The fraction of trees with a trunk diameter of at least 15 inches that have a trunk diameter of less than 25 inches is 37/52.
To calculate the required fraction, first identify the subset of trees that have a trunk diameter of at least 15 inches. Summing the frequencies of the intervals 15–19, 20–24, 25–29, and 30–34 gives 15 + 22 + 11 + 4 = 52 trees. Next, identify how many of these 52 trees have a trunk diameter of less than 25 inches. These are the trees in the intervals 15–19 and 20–24, giving 15 + 22 = 37 trees. Therefore, the fraction of trees meeting the condition is 37/52.

Adım Adım Çözüm

1
Determine the total number of trees satisfying the condition 'at least 15 inches'.
Sum the frequencies for intervals 15–19, 20–24, 25–29, and 30–34: 15+22+11+4=5215 + 22 + 11 + 4 = 52 trees.
This establishes the restricted denominator (base group) specified by the conditional statement.
2
Determine the number of trees within this group that satisfy the condition 'less than 25 inches'.
Sum the frequencies for intervals 15–19 and 20–24: 15+22=3715 + 22 = 37 trees.
These are the target intervals that are both at least 15 inches and strictly less than 25 inches.
3
Calculate the required fraction.
3752\frac{37}{52}.
Divide the target count by the conditional total count to form the part-to-whole ratio.

Anahtar Kavram

Conditional relative frequencies and part-to-whole calculations from grouped data tables.
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