Twenty middle-aged men with glucose readings between 90 milligrams per deciliter and 120 milligrams per deciliter of blood were selected randomly from the population of similar male patients at a large local hospital. Ten of the 20 men were assigned randomly to group A and received a placebo. The other 10 men were assigned to group B and received a new glucose drug. After two months, posttreatment glucose readings were taken for all 20 men and were compared with pretreatment readings. The reduction in glucose level (Pretreatment reading − Posttreatment reading) for each man in the study is shown here. Group A (placebo) reduction (in milligrams per deciliter): 12, 8, 17, 7, 20, 2, 5, 9, 12, 6 Group B (glucose drug) reduction (in milligrams per deciliter): 29, 31, 13, 19, 21, 5, 24, 12, 8, 21 Part A: Do the data provide convincing evidence, at the α = 0.02 level, that the glucose drug is effective in producing a reduction in mean glucose level?  Part B: Create and interpret a 98% confidence interval for the difference in the placebo and the new drug.

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 + ...