A boat capsized and sank in a lake. Based on an assumption of the mean weight of 130​lb, After the boat​ sank, the assumed mean weight for similar boats was changed from 130 lb to 171 lb. Complete parts a and b below. a. Assume that a similar boat is loaded with 70 ​passengers, and assume that the weights of people are normally distributed with a mean of 182.7 lb and a standard deviation of 40.4 lb. Find the probability that the boat is overloaded because the 70 passengers have a mean weight greater than 130 lb. The probability is _____.​(Round to four decimal places as​ needed.) b. The boat was later rated to carry only 14 ​passengers, and the load limit was changed to 2,394 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 171 ​(so that their total weight is greater than the maximum capacity of 2,394 ​lb). The probability is ______​(Round to four decimal places as​ needed.)

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...