Before every flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft. The aircraft can carry 39 passengers, and a flight has and baggage that allows for a total passenger load of 6,396 lb. The pilot sees that the plane is full and all passengers are men. The aircraft will be overloaded if the mean weight of the 6,396 lb = 164 lb. What is the probability that the aircraft is overloaded? Should the pilot take any action to correct for an overloaded aircraft? Assume that 39 weights of men are normally distributed with a mean of 182.4 lb and a standard deviation of 39.7. passengers is greater than The probability is approximately. (Round to four decimal places as needed.)

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**Probability of Aircraft Overloading** 

In ensuring flight safety, a pilot must confirm that the total weight does not exceed the aircraft's maximum allowable load. An aircraft with a capacity of 39 passengers, including fuel and baggage, allows for a total passenger load of 6,396 lbs. If the plane is at full capacity with male passengers, overloading occurs when the mean weight per passenger exceeds \( \frac{6,396}{39} = 164 \) lbs.

**Question:**
What is the probability that the aircraft is overloaded? Should the pilot take corrective action? Assume the weights of men are normally distributed with a mean of 182.4 lbs and a standard deviation of 39.7 lbs.

**Solution Approach:**
1. Standardize the problem using the normal distribution model.
2. Calculate the probability that the average weight is greater than 164 lbs.
3. Determine if any corrective actions are necessary based on the probability calculated.

**Calculation:**
- Mean (\(\mu\)) = 182.4 lbs
- Standard deviation (\(\sigma\)) = 39.7 lbs
- Desired average weight threshold = 164 lbs

The probability is approximately \[ \_\_\_\_ \].
(Round to four decimal places as needed.)

**Action:**
Assess whether this probability warrants action to prevent overloading.

*Note: Detailed calculations involving Z-scores and normal distribution tables might be required to find the exact probability value.*
Transcribed Image Text:**Probability of Aircraft Overloading** In ensuring flight safety, a pilot must confirm that the total weight does not exceed the aircraft's maximum allowable load. An aircraft with a capacity of 39 passengers, including fuel and baggage, allows for a total passenger load of 6,396 lbs. If the plane is at full capacity with male passengers, overloading occurs when the mean weight per passenger exceeds \( \frac{6,396}{39} = 164 \) lbs. **Question:** What is the probability that the aircraft is overloaded? Should the pilot take corrective action? Assume the weights of men are normally distributed with a mean of 182.4 lbs and a standard deviation of 39.7 lbs. **Solution Approach:** 1. Standardize the problem using the normal distribution model. 2. Calculate the probability that the average weight is greater than 164 lbs. 3. Determine if any corrective actions are necessary based on the probability calculated. **Calculation:** - Mean (\(\mu\)) = 182.4 lbs - Standard deviation (\(\sigma\)) = 39.7 lbs - Desired average weight threshold = 164 lbs The probability is approximately \[ \_\_\_\_ \]. (Round to four decimal places as needed.) **Action:** Assess whether this probability warrants action to prevent overloading. *Note: Detailed calculations involving Z-scores and normal distribution tables might be required to find the exact probability value.*
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