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The image shows a question related to the analysis of a simple random sample of flight durations for a space shuttle program, where you need to compute the range, variance, and standard deviation. Here's a breakdown of the components visible in the image:

1. **Data set** (in hours):  
   74, 92, 239, 193, 169, 267, 191, 372, 253, 232, 386, 332, 239, 241, 0

2. **Calculations:**
   - **Range**: The range is calculated as the difference between the maximum and minimum values in the data set.
   - **Variance**: The variance is given as **11267.1** hours².
   - **Standard Deviation**: The standard deviation is **106.1** hours.

3. **Question about Unusual Value**:
   The prompt asks whether the lowest duration (0 hours) is unusual based on standard deviation. The correct answer marked is **C**: 
   "No, because it is within two standard deviations of the mean."

### Let's break down the missing calculation:

- **Range Calculation**:  
   \[ \text{Range} = \text{Maximum Value} - \text{Minimum Value} \]
   Using the dataset:
   \[
   \text{Range} = 386 - 0 = 386 \text{ hours}
   \]
  
### Key Conclusion:
The value 0 is considered **not unusual** because it falls within two standard deviations of the mean.

Would you like more details on the computation of variance and standard deviation? Or do you have any specific follow-up questions?

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Here are 5 related questions to expand your understanding:
1. How is the mean calculated for this data set, and how does it relate to variance?
2. Can you explain why values beyond two standard deviations are typically considered unusual?
3. What is the impact of the value 0 on the variance and standard deviation of this data set?
4. How does removing the outlier (if any) affect the spread of the data (range, variance)?
5. Can you explore how skewness in the data affects the interpretation of standard deviation?

**Tip**: The square root of variance always gives the standard deviation. Hence, calculating one from the other is straightforward.

Listed below are the durations (in hours) of a simple random sample of all flights of a space shuttle program. Find the range, variance, and standard deviation for the sample data. Is the lowest duration time unusual? Why or why not?

Analyze the sample of space shuttle flight durations, including calculations of range, variance, and standard deviation. Explore whether the lowest duration is unusual based on standard deviation. The dataset includes values such as 74, 92, and 239 hours, and key calculations like a variance of 11267.1 and a standard deviation of 106.1.

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