Caffeinated Beverages and Diabetes. The objective of the article, “Caffeinated and Caffeine-free Beverages and Risk of Type 2 Diabetes” (American Journal of Clinical Nutrition, Vol. 97, No. 1, pp. 155–166) by S. Bhupathiraju et al., was to examine the association between caffeinated beverages and type 2 diabetes risk. The mean and standard deviation of the body mass index (BMI) for a sample of 10,215 women who drink at least one caffeinated carbonated beverage a day are 25.7 and 5.3, respectively. a. At least how many women in the sample have a BMI of between 15.1 and 36.3? b. Fill in the blanks: At least 89% of the women in the sample have BMIs between ____and ______

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 + ...