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Zorluk: OrtaFrequency Distributions, Tables, and Grouped Data

The table below shows the frequency distribution of order fulfillment times, in minutes, for a sample of 5050 online orders processed at a warehouse facility.

Fulfillment Time tt (minutes)Number of Orders
10≤t<2010 \le t < 201010
20≤t<3020 \le t < 301515
30≤t<4030 \le t < 401515
40≤t<5040 \le t < 5077
50≤t<6050 \le t < 6033

An order is categorized as "delayed" if its fulfillment time is at least 3030 minutes. What percentage of all orders that took at least 2020 minutes to fulfill were categorized as "delayed"?

  1. A
    30.0%
  2. B
    37.5%
  3. C
    50.0%
  4. 62.5%Cevap
  5. E
    75.0%

Cevap

62.5%
To find the required percentage, first determine the subset of orders that took at least 2020 minutes to fulfill. Summing the frequencies of the last four intervals (20≤t<3020 \le t < 30, 30≤t<4030 \le t < 40, 40≤t<5040 \le t < 50, and 50≤t<6050 \le t < 60) yields 15+15+7+3=4015 + 15 + 7 + 3 = 40 orders. Next, find how many of these orders took at least 3030 minutes to fulfill (the delayed orders), which corresponds to the last three intervals: 15+7+3=2515 + 7 + 3 = 25 orders. Finally, calculate the percentage: 2540×100%=62.5%\frac{25}{40} \times 100\% = 62.5\%.

Adım Adım Çözüm

1
Identify the sub-population that defines the base (denominator) of the percentage.
The target base is all orders with fulfillment times of at least 2020 minutes (t≥20t \ge 20). Summing the frequencies for intervals 20≤t<3020 \le t < 30, 30≤t<4030 \le t < 40, 40≤t<5040 \le t < 50, and 50≤t<6050 \le t < 60 gives 15+15+7+3=4015 + 15 + 7 + 3 = 40 orders.
The question specifically asks 'of all orders that took at least 20 minutes to fulfill', establishing a conditional base.
2
Determine the number of orders meeting the numerator condition.
The delayed orders are those taking at least 3030 minutes (t≥30t \ge 30). Summing frequencies for 30≤t<4030 \le t < 40, 40≤t<5040 \le t < 50, and 50≤t<6050 \le t < 60 gives 15+7+3=2515 + 7 + 3 = 25 orders.
Orders in all intervals starting from 3030 minutes onwards qualify as delayed.
3
Compute the required percentage.
(2540)×100%=0.625×100%=62.5%\left(\frac{25}{40}\right) \times 100\% = 0.625 \times 100\% = 62.5\%.
Dividing the target count by the restricted base count gives the conditional proportion.

Anahtar Kavram

Grouped Frequency Distributions and Conditional Percentages
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Frequency Distributions, Tables, and Grouped Data

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