The united states coast guard assumes the mean weight of passengers in commercial boats is 185 pounds with a standard deviation of 26.7 pounds. what is the probability that a random sample of 36 passengers has a mean weight between 175 and 195 pounds?

To find the probability that a random sample of 36 passengers has a mean weight between 175 and 195 pounds, we can use the Central Limit Theorem and z-scores. Use the z-scores to find the cumulative probabilities and subtract the lower boundary probability from the upper boundary probability.

Explanation:

To find the probability that a random sample of 36 passengers has a mean weight between 175 and 195 pounds, we can use the Central Limit Theorem. Since we have a large sample size (n≥30), we can assume that the sample mean follows a normal distribution. The mean weight of 185 pounds and standard deviation of 26.7 pounds will be used to calculate the z-scores of the lower and upper boundaries. We can then use a standard normal table or calculator to find the probability.

Calculate the z-scores for the lower and upper boundaries: z1 = (175 - 185) / (26.7 / √36) and z2 = (195 - 185) / (26.7 / √36).
Look up the cumulative probability for each z-score in a standard normal table or use a calculator to find the area under the normal distribution curve between the two z-scores.
Subtract the cumulative probability of the lower boundary from the cumulative probability of the upper boundary to find the probability that a random sample of 36 passengers has a mean weight between 175 and 195 pounds.

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