In response to the increasing weight of airline passengers, the Federal Aviation Administration in 2003 told airlines to assume that passengers average 190 pounds in the summer, including clothing and carry-on baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation is 35 pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-Normal.
1. A commuter plane carries 19 passengers. What is the approximate probability that the total weight of the passengers exceeds 4000 pounds? Use the four-step process and Table A to guide your work.
(Hint: To apply the central limit theorem, restate the problem in terms of the mean weight.)
What four step process? Also, we do not have access to Table A.
To solve this problem, we can use the Central Limit Theorem (CLT) to approximate the distribution of the total weight of the passengers.
Here are the four steps we can follow:
Step 1: Identify the Distribution
- Since the weights are not normally distributed and there is no information on the actual distribution shape, we can approximate it as a normal distribution based on the Central Limit Theorem.
Step 2: Define the Parameter
- We are interested in the total weight of 19 passengers, so our parameter of interest is the mean weight of the passengers.
Step 3: Formulate the Hypotheses
- In this case, we are interested in the probability that the total weight of the passengers exceeds 4000 pounds. So our hypotheses are:
- Null hypothesis (H0): The mean weight of the passengers is less than or equal to 4000 pounds.
- Alternative hypothesis (Ha): The mean weight of the passengers exceeds 4000 pounds.
Step 4: Calculate the Probability
- To calculate the approximate probability, we will need to calculate the z-score and then use the Standard Normal Distribution (Table A) to find the corresponding probability.
Here's how we can proceed:
1. Calculate the mean weight (μ) of the passengers:
μ = 190 pounds (given)
2. Calculate the standard deviation (σ) of the mean weight using the given standard deviation (σ) and the number of passengers (n):
σ_mean = σ / √(n)
σ_mean = 35 / √19
3. Calculate the z-score:
z = (Total weight - n * μ) / (σ_mean)
z = (4000 - 19 * 190) / (35 / √19)
4. Using Table A or a statistical calculator/tool, find the probability corresponding to the calculated z-score.
The probability will be the area under the normal curve to the right of the z-score.
Note: Table A provides the cumulative probabilities to the left of the z-score. Since we need the probability to the right, we can subtract the cumulative probability from 1.
By following these steps, you should be able to determine the approximate probability that the total weight of the passengers exceeds 4000 pounds.