Question 1204046
**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.