Students in a statistics class answered a quiz question and the time it took each to complete it was recorded. The results are summarized in the following frequency distribution. Length of Time (in minutes) Number 0 up to 2 3 2 up to 4 4 up to 6 20 6 up to 10 8. What is the mean (in minutes)?

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### Analysis of Quiz Completion Times

Students in a statistics class answered a quiz question and the time it took each to complete it was recorded. The results are summarized in the following frequency distribution:

| **Length of Time (in minutes)** | **Number** |
|---------------------------------|------------|
| 0 up to 2                       | 3          |
| 2 up to 4                       | 6          |
| 4 up to 6                       | 20         |
| 6 up to 10                      | 8          |

**Question:** What is the mean (in minutes)?

To calculate the mean, we need to find the midpoint for each class interval, multiply by the frequency for that interval, sum these products, and then divide by the total number of observations.

Steps to calculate the mean:
1. Determine the midpoint (xᵢ) for each interval.
2. Multiply each midpoint by the corresponding frequency (fᵢ).

Let's calculate step-by-step:

1. Midpoints:
   - For 0 up to 2: \((0 + 2) / 2 = 1\)
   - For 2 up to 4: \((2 + 4) / 2 = 3\)
   - For 4 up to 6: \((4 + 6) / 2 = 5\)
   - For 6 up to 10: \((6 + 10) / 2 = 8\)

2. Frequency times Midpoint:
   - \(1 \times 3 = 3\)
   - \(3 \times 6 = 18\)
   - \(5 \times 20 = 100\)
   - \(8 \times 8 = 64\)

3. Sum of these products: 
   \[3 + 18 + 100 + 64 = 185\]

4. Total number of observations:
   \[3 + 6 + 20 + 8 = 37\]

5. Calculating the mean:
   \[
   \text{Mean} = \frac{\sum (fᵢ \times xᵢ)}{\sum fᵢ} = \frac{185}{37} \approx 5 \text{ minutes}
   \]

Thus, the mean time taken to complete the quiz is approximately **5 minutes**.
Transcribed Image Text:### Analysis of Quiz Completion Times Students in a statistics class answered a quiz question and the time it took each to complete it was recorded. The results are summarized in the following frequency distribution: | **Length of Time (in minutes)** | **Number** | |---------------------------------|------------| | 0 up to 2 | 3 | | 2 up to 4 | 6 | | 4 up to 6 | 20 | | 6 up to 10 | 8 | **Question:** What is the mean (in minutes)? To calculate the mean, we need to find the midpoint for each class interval, multiply by the frequency for that interval, sum these products, and then divide by the total number of observations. Steps to calculate the mean: 1. Determine the midpoint (xᵢ) for each interval. 2. Multiply each midpoint by the corresponding frequency (fᵢ). Let's calculate step-by-step: 1. Midpoints: - For 0 up to 2: \((0 + 2) / 2 = 1\) - For 2 up to 4: \((2 + 4) / 2 = 3\) - For 4 up to 6: \((4 + 6) / 2 = 5\) - For 6 up to 10: \((6 + 10) / 2 = 8\) 2. Frequency times Midpoint: - \(1 \times 3 = 3\) - \(3 \times 6 = 18\) - \(5 \times 20 = 100\) - \(8 \times 8 = 64\) 3. Sum of these products: \[3 + 18 + 100 + 64 = 185\] 4. Total number of observations: \[3 + 6 + 20 + 8 = 37\] 5. Calculating the mean: \[ \text{Mean} = \frac{\sum (fᵢ \times xᵢ)}{\sum fᵢ} = \frac{185}{37} \approx 5 \text{ minutes} \] Thus, the mean time taken to complete the quiz is approximately **5 minutes**.
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