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Assume that a similar boat is loaded with 5050 passengers, and the weights of people are normally distributed with a mean of 176.6 lb and a standard deviation of 38.2 lb.

Find the probability that the boat is overloaded because the 5050 passengers have a mean weight greater than 139 lb.

Answer :

To determine the probability that the boat is overloaded because the 5050 passengers have a mean weight greater than 139 lb, we need to apply the concepts of normal distribution and the Central Limit Theorem.

Step-by-Step Explanation:

  1. Identify the Given Information:

    • Mean weight of a single passenger: [tex]\mu = 176.6[/tex] lb
    • Standard deviation of a single passenger's weight: [tex]\sigma = 38.2[/tex] lb
    • Number of passengers [tex]n = 5050[/tex]
    • We want to find the probability that the mean weight of 5050 passengers is greater than 139 lb.
  2. Calculate the Standard Error of the Mean (SEM):

    The standard error of the mean is calculated using the formula:
    [tex]\text{SEM} = \frac{\sigma}{\sqrt{n}}[/tex]
    Substitute the given values:
    [tex]\text{SEM} = \frac{38.2}{\sqrt{5050}} \approx 0.537[/tex]

  3. Determine the Z-score:

    We need to find the Z-score to determine how many standard errors the sample mean of 139 lb is from the actual mean of 176.6 lb. The formula for the Z-score is:
    [tex]Z = \frac{\bar{x} - \mu}{\text{SEM}}[/tex]
    where [tex]\bar{x} = 139[/tex] is the sample mean.
    [tex]Z = \frac{139 - 176.6}{0.537} \approx -69.86[/tex]

  4. Find the Probability:

    The Z-score corresponds to a point on the standard normal distribution. Since the Z-score is negative and large in magnitude, it implies that the probability of the mean being greater than 139 lb is extremely high, essentially 1.

    In practice, a Z-score so far from the mean indicates that the event of having a mean weight less than 139 lb is nearly impossible, given the provided mean and standard deviation.

In conclusion, the probability that the boat is overloaded because the 5050 passengers have a mean weight greater than 139 lb is practically 100%. This means the average weight of passengers should comfortably exceed 139 lb under normal circumstances.

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