To calculate the standard deviation, we first need to determine the mean (average) of the values given and then use that to find the variance and standard deviation.
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Calculate the mean:
Using the values and total observations:
- For 3 miles: 2 days
- For 4 miles: 3 days
- For 5 miles: 2 days
The total number of observations (days) is: \[ \text{Total Days} = 2 + 3 + 2 = 7 \]
Now, calculate the weighted average (mean): \[ \text{Mean} = \frac{(3 \times 2) + (4 \times 3) + (5 \times 2)}{7} \] \[ = \frac{(6) + (12) + (10)}{7} = \frac{28}{7} = 4 \]
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Calculate the variance:
The variance is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2 \cdot f_i}{N} \] where \(x_i\) are the values, \(\mu\) is the mean, \(f_i\) is the frequency (total observations), and \(N\) is the total observations.
Now calculate \((x_i - \mu)^2\) for each value:
- For 3 miles: \[ (3 - 4)^2 \cdot 2 = (-1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
- For 4 miles: \[ (4 - 4)^2 \cdot 3 = (0)^2 \cdot 3 = 0 \cdot 3 = 0 \]
- For 5 miles: \[ (5 - 4)^2 \cdot 2 = (1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
Now sum these values: \[ \text{Total} = 2 + 0 + 2 = 4 \]
Now calculate the variance: \[ \sigma^2 = \frac{4}{7} \approx 0.57142857 \]
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Calculate the standard deviation: \[ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{4}{7}} \approx 0.75 \]
Rounding this to the nearest hundredth gives approximately \(0.75\).
Based on the responses given:
- 0.45
- 0.67
- 0.76
- 0.82
The closest response to our calculated standard deviation \(0.75\) is 0.76.