Final answer:
The mean of the depth is 13.75 cm and the variance is 33.29 cm^2. The CDF of the depth is a linear function. The probability that the observed depth is at most 10 is 0.1939. The probability that the observed depth is between 10 and 15 is 0.2903. The probability that the observed depth is within one standard deviation of the mean is 0.6378. The probability that the observed depth is within two standard deviations of the mean is 1.
Step-by-step explanation:
The mean of a uniform distribution can be calculated as the average of the two endpoints of the interval, which in this case is (6.5 + 21) / 2 = 13.75 cm.
The variance of a uniform distribution can be calculated using the formula: variance = ((b - a)^2) / 12, where 'a' and 'b' are the endpoints of the interval. In this case, the variance = ((21 - 6.5)^2) / 12 = 33.29 cm^2 (rounded to two decimal places).
The cumulative distribution function (CDF) of a uniform distribution is a linear function. For X < 6.5, the probability is 0. For 6.5 < X < 21, the probability is (X - 6.5) / (21 - 6.5). For X > 21, the probability is 1.
The probability that the observed depth is at most 10 can be calculated as (10 - 6.5) / (21 - 6.5) = 0.1939 (rounded to four decimal places).
The probability that the observed depth is between 10 and 15 can be calculated as (15 - 6.5) / (21 - 6.5) - (10 - 6.5) / (21 - 6.5) = 0.2903 (rounded to four decimal places).
The probability that the observed depth is within one standard deviation of the mean can be calculated as (13.75 + 6.5) / (21 - 6.5) = 0.6378 (rounded to four decimal places).
The probability that the observed depth is within two standard deviations of the mean can be calculated as (13.75 + 2*(21 - 6.5)) / (21 - 6.5) = 1 (rounded to four decimal places).