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?

7373   

9595   

235235   

193193   

163163   

265265   

200200   

373373   

254254   

234234   

381381   

330330   

228228   

245245   

00 
  

To calculate the **range**, **variance**, and **standard deviation** for the given sample data, let's first organize the sample and define the steps:

### Data Set (in hours):
73, 95, 235, 193, 163, 265, 200, 373, 254, 234, 381, 330, 228, 245, 0

#### Steps:
1. **Range**: This is the difference between the maximum and minimum values in the sample.
   
2. **Variance**: This measures how far the numbers in the data set are from the mean (average). It is calculated as:
   \[
   \text{Variance} = \frac{\sum{(x_i - \bar{x})^2}}{n-1}
   \]
   where \( \bar{x} \) is the sample mean, and \( n \) is the number of data points.

3. **Standard Deviation**: This is the square root of the variance.

I'll calculate these values for you.### Results:
1. **Range**: The range is \( 381 \) hours.
2. **Variance**: The variance is approximately \( 11119.21 \) hours².
3. **Standard Deviation**: The standard deviation is approximately \( 105.45 \) hours.

### Is the lowest duration (0 hours) unusual?
To determine whether 0 hours is unusual, we check if it lies far from the mean using the empirical rule (which states that most data in a normal distribution falls within 2 or 3 standard deviations of the mean). Let's check:

- Mean \( (\bar{x}) \): \( \approx 224.07 \) hours.
- The lowest value (0) is approximately \( 224.07 \) hours away from the mean.
- This is about \( \frac{224.07}{105.45} \approx 2.12 \) standard deviations below the mean.

Since the value of 0 hours lies slightly more than 2 standard deviations below the mean, it could be considered unusual, though not extremely rare.

Would you like further details or explanations?

Here are 5 follow-up questions to consider:
1. How do we interpret the standard deviation in practical terms?
2. What are other methods to determine if a value is unusual in a dataset?
3. How would the range and variance change if we remove the lowest value?
4. What is the impact of outliers on the variance and standard deviation?
5. How do different data distributions affect the empirical rule?

**Tip:** Always check for outliers, as they can significantly impact variance and standard deviation.

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?

Find the range, variance, and standard deviation of space shuttle flight durations, including an analysis of whether the lowest flight time is unusual. This problem focuses on statistical concepts like range, variance, and standard deviation, perfect for high school students in Grades 10-12.

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