A study seeks to estimate the difference in the mean fuel economy (measured in miles per gallon) for vehicles under two treatments: driving with underinflated tires versus driving with properly inflated tires. To quantify this difference, the manufacturer randomly selects 12 cars of the same make and model from the assembly line and then randomly assigns six of the cars to be driven 500 miles with underinflated tires and the other six cars to be driven 500 miles with properly inflated tires. A 90% confidence interval for the true difference in the mean fuel economy for cars of this make and model driven with underinflated tires versus properly inflated tires is –3.339 mpg to –0.585 mpg. Interpret this interval. The manufacturer can be 95% confident that the interval from –3.339 to –0.585 captures the sample mean difference in fuel economy (underinflated – properly inflated). The manufacturer can be 95% confident that the interval from –3.339 to –0.585 captures the true mean difference in fuel economy (underinflated – properly inflated). The manufacturer can be 95% confident that a randomly selected car with underinflated tires will get between 3.339 and 0.585 fewer miles per gallon than a randomly selected car with properly inflated tires. The manufacturer can be 95% confident that a randomly selected group of six cars with underinflated tires will get between 3.339 and 0.585 fewer miles per gallon than a randomly selected group of six cars with properly inflated tires.

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 + ...