Final answer:
To find the probability of the mean weight of 10 adults being over 200 pounds, calculate the standard error of the mean, determine the Z-score for a sample mean of 200 pounds, and use the Z-table to find the corresponding probability.
Step-by-step explanation:
To determine the probability that the mean weight of a random sample of 10 adults is over 200 pounds, we first need to identify the distribution of the sample mean. Since the average adult weighs 175 pounds with a standard deviation of 25 pounds and the sample size n=10, we can use the Central Limit Theorem (CLT) which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough, which we will assume is the case here. We then calculate the standard error of the mean (SEM) which is the standard deviation divided by the square root of the sample size: SEM = 25 / sqrt(10).
Next, we calculate the Z-score for a sample mean of 200 pounds using the formula Z = (X - μ) / SEM, where X is the sample mean we are interested in, and μ is the population mean. Substituting the values, we get Z = (200 - 175) / (25 / sqrt(10)). After computing this, we look up this Z-score in the standard normal distribution table (Z-table) to find the probability that Z is less than our calculated value. We then subtract this probability from 1 to find the probability of Z being greater than the calculated value, which is our desired probability of the mean weight being over 200 pounds.
Using a calculator or Z-table, we find the probability that corresponds to our Z-score and calculate the final probability, which should be rounded to four decimal places as requested.