A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 102.7-cm and a standard deviation of 1.2-cm. For shipment, 45 steel rods are bundled together. Round all answers to four decimal places if necessary. a. What is the distribution of X? X - N( b. What is the distribution of x? x - N( c. For a single randomly selected steel rod, find the probability that the length is between 102.4-cm and 102.7-cm. d. For a bundled of 45 rods, find the probability that the average length is between 102.4-cm and 102.7-cm. e. For part d), is the assumption of normal necessary? Yes No

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 + ...