Confidence Intervals – Dr. Richards is researching the effectiveness of several new, organic types of fertilizer that are also good for the environment. In her sampling, she finds that the mean improvement in crop yields from three different fertilizers are all 23%. However, the 95% confidence intervals of the three fertilizers differ as follows:   Fertilizer Mean Crop Yield Improvement 95% Confidence Interval for Yield Improvement 1 23% 22% - 24% 2 23% 20% - 26% 3 23% 18% - 28%   Explain what this it means to have the same sample means but different confidence intervals. What are the implications for Dr. Richards research?

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 + ...