A research study investigated the composition of wines from the Rhine and Moselle wine regions of Germany. One component measured was the concentration of i-Butyl alcohol. The results of the study are listed in the following table: Region n i-Butyl alcohol (mg/100ml) Rhine 9 ̄x =5.32,s =2.24 Moselle 14 ̄x =3.62,s =0.97 (a) Assume that the underlying population variances from the two regions are identical. At the 0.05 level of significance, test the null hypothesis that the mean concentration of i-Butyl alcohol is the same for wines from the Rhine and Moselle regions. Construct a 95% confidence interval (b) Now assume that the underlying population variances from the two regions differ. At the 0.05 level of significance, test the null hypothesis that the mean concentration of i-Butyl alcohol is the same for wines from the Rhine and Moselle regions, construct a 95% confidence interval (c) If you were to provide a report to the investigators which of the above results would you include and why?

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 + ...