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Question 1204046: The amount of caffeine in a sample of 250ml servings of brewed coffee is summarized in the table below:
Caffeine (mg) Number of cups
60 < 80 : 1
80 < 100 : 12
100 < 120 : 25
120 < 140 : 10
140 < 160 : 2
2.1 Calculate the average caffeine content of the 250ml cup
2.2 Calculate the modal caffeine content of the 250ml cup.
2.3 Calculate the median caffeine content of the 250ml cup.
2.4 Calculate the standard deviation of the content of the 250ml cup
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **2.1 Calculate the average caffeine content of the 250ml cup**
1. **Find the midpoint of each caffeine range:**
* 60-80 mg: 70 mg
* 80-100 mg: 90 mg
* 100-120 mg: 110 mg
* 120-140 mg: 130 mg
* 140-160 mg: 150 mg
2. **Calculate the weighted average:**
* Average = [(70 * 1) + (90 * 12) + (110 * 25) + (130 * 10) + (150 * 2)] / (1 + 12 + 25 + 10 + 2)
* Average = [70 + 1080 + 2750 + 1300 + 300] / 50
* Average = 5500 / 50
* Average = 110 mg
**Therefore, the average caffeine content of the 250ml cup is 110 mg.**
**2.2 Calculate the modal caffeine content of the 250ml cup**
* The mode is the value that occurs most frequently.
* In this case, the 100-120 mg range has the highest frequency (25 cups).
**Therefore, the modal caffeine content of the 250ml cup is the 100-120 mg range.**
**2.3 Calculate the median caffeine content of the 250ml cup**
* The median is the middle value when the data is arranged in order.
* **Find the cumulative frequency:**
* 60-80 mg: 1 cup
* 80-100 mg: 13 cups (1 + 12)
* 100-120 mg: 38 cups (13 + 25)
* 120-140 mg: 48 cups (38 + 10)
* 140-160 mg: 50 cups (48 + 2)
* **Identify the median class:**
* The median falls within the 100-120 mg range as it contains the 25th and 26th observations.
* **Estimate the median:**
* Since the median class contains 25 observations, the median is likely to be closer to the upper limit of the class.
* A rough estimate of the median could be around 110 mg.
**2.4 Calculate the standard deviation of the content of the 250ml cup**
1. **Calculate the deviations from the mean:**
* 60-80 mg: 70 - 110 = -40
* 80-100 mg: 90 - 110 = -20
* 100-120 mg: 110 - 110 = 0
* 120-140 mg: 130 - 110 = 20
* 140-160 mg: 150 - 110 = 40
2. **Square the deviations:**
* (-40)² = 1600
* (-20)² = 400
* 0² = 0
* 20² = 400
* 40² = 1600
3. **Multiply the squared deviations by their frequencies:**
* 1600 * 1 = 1600
* 400 * 12 = 4800
* 0 * 25 = 0
* 400 * 10 = 4000
* 1600 * 2 = 3200
4. **Sum the products:**
* 1600 + 4800 + 0 + 4000 + 3200 = 13600
5. **Divide the sum by the total number of observations (50):**
* 13600 / 50 = 272
6. **Take the square root of the result:**
* √272 ≈ 16.49
**Therefore, the standard deviation of the caffeine content of the 250ml cup is approximately 16.49 mg.**
**Note:** These calculations provide estimates of the mean, median, and standard deviation. More precise values could be obtained with the exact caffeine content of each cup.
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