To find the mean and standard deviation of the number of passengers who do not show up for their flight, we can model the situation using a binomial distribution.
Let:
- \( n = 185 \) (the total number of passengers),
- \( p = 0.0437 \) (the probability that a passenger does not show up).
Mean
The mean \( \mu \) of a binomial distribution is given by the formula: \[ \mu = n \cdot p \] Substituting the values, we have: \[ \mu = 185 \cdot 0.0437 = 8.0945 \] So the mean number of passengers who do not show up is approximately 8.09.
Standard Deviation
The standard deviation \( \sigma \) of a binomial distribution is given by the formula: \[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \] Calculating \( 1 - p \): \[ 1 - p = 1 - 0.0437 = 0.9563 \]
Now substituting the values into the standard deviation formula: \[ \sigma = \sqrt{185 \cdot 0.0437 \cdot 0.9563} \] Calculating this step-by-step:
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Calculate \( n \cdot p \cdot (1 - p) \): \[ 185 \cdot 0.0437 \cdot 0.9563 \approx 185 \cdot 0.0418 \approx 7.733 \]
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Now, take the square root: \[ \sigma \approx \sqrt{7.733} \approx 2.78 \]
Summary
- Mean number of passengers who do not show up: approximately 8.09
- Standard deviation of the number of passengers who do not show up: approximately 2.78.