The Department of Transportation (DoT) is concerned about the high speed of vehicles traveling down a dangerous stretch of an interstate highway with a posted speed limit of 55 mph. Part I An extensive study has revealed that the speed of vehicles traveling down the dangerous stretch is normally distributed with a mean speed of 65.3 mph and a standard deviation of 6.8 mph. (a) What is the probability that a random vehicle travels at a speed in excess of 20 mph of the posted speed limit? (Round your answer to three decimal places.)   (b) What is the probability that a random vehicle travels slower than 5 mph under the posted speed limit? (Round your answer to three decimal places.)   (c) What is the probability that a random vehicle travels at a speed that is between the posted speed limit and 10 mph in excess of it? (Round your answer to three decimal places.)   (d) What speed corresponds to the 90th percentile? (Round your answer to one decimal place.)  mph Part II The Department of Transportation (DoT) collects the speeds of a random sample of 95 vehicles traveling down the dangerous stretch of highway. (e) How many of the 95 vehicles would the DoT expect to travel at a speed that is between the posted speed limit and 10 mph in excess of it? (Round your answer to the nearest integer.)  vehicles (f) Show the sampling distribution of the sample mean speed.               (g) What is the probability that the sample mean speed falls within 1 mph of the population mean? (Round your answer to three decimal place.)   (h) What is the probability that the sample mean speed is in excess of 66.4 mph? (Round your answer to three decimal place.)   Part III The Department of Transportation (DoT) decides to seek the help of state police, who respond by heavily patrolling the dangerous stretch of highway as a deterrent. When the DoT monitors again the speed of vehicles, it discovers that the speed distribution is still normal, but the mean speed has dropped to 55.3 mph, and the standard deviation to 3.4 mph. (i) What is now the probability that a random vehicle travels at a speed that is between the posted speed limit and 10 mph in excess of it? (Round your answer to three decimal place.)   (j) Using the newly reduced mean speed and standard deviation, what speed now corresponds to the 90th percentile? (Round your answer to one decimal place.)  mph (k) Using the newly reduced mean speed and standard deviation, how many of the 95 vehicles traveling down the dangerous stretch of highway would the DoT now expect to travel at a speed that is between the posted speed limit and 10 mph in excess of it? (Round your answer to the nearest integer.)  vehicles

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 + ...