Final answer:
The probability that the sample mean is less than 2.51 inches in a uniform distribution (U(1.5, 4.5)) is calculated as the proportion of the interval lying to the left of 2.51 inches, resulting in approximately 33.67%.
Step-by-step explanation:
To calculate the probability that the sample mean is less than 2.51 inches, given that the sample mean is 2.50 inches and the standard deviation is 0.8302, one would typically use the Z-score formula. However, since we know the distribution is uniform (X ~ U(1.5, 4.5)), we calculate the probability directly from the uniform distribution properties.
The uniform distribution is defined such that every outcome between the minimum value (1.5 inches) and the maximum value (4.5 inches) is equally likely. To find the probability that the sample mean is less than 2.51 inches, we can think of this as finding what proportion of the total length of the interval (from 1.5 to 4.5 inches) lies to the left of 2.51 inches.
The calculation is as follows: P(X < 2.51) = (2.51 - 1.5) / (4.5 - 1.5) = 1.01 / 3 = 0.3367, approximately. Therefore, the probability that the sample mean is less than 2.51 inches is roughly 33.67%.