Problem 10.1 (small sample). Suppose that we observe the following lactic acid concentrations in cheese: (G. P. McCabe, L. McCabe, and A. Miller, Analysis of taste and chemical composition of cheddar cheese, 1982-83 experiments, CSIRO Division of Mathematics and Statistics Consulting Report VT85/6): X1 = 0.86, X2 = 1.53, X3 = 1.57, X4 = 1.81, X5 = 0.99, X6 = 1.09, X7 = 1.29, X8 = 1.78, X9 = 1.29, X10 = 1.58. Assuming that these were independent observations, find the 90%, the 95%, as well as the 99% confidence intervals for , the unknown mean lactic acid concentration. Hint: When you are in a situation like this (small sample size) you may assume that the entire population is Normally distributed even if it is not specified by the problem, so to be able to use the table of critical values for Student's t distribution.

Problem 10.2 (estimation of , with known variance). With reference to the previous problem, suppose that we have some extra information (quite possibly, from a previous study): we know that the true standard deviation of lactic acid concentration in cheese is = 0.340. (a) How do the answers of the previous problem change? I.e. what are the 90%, the 95%, as well as the 99% confidence intervals for ? (b) What sample size n do we need to have a 95% margin of error of at most 0.05? Remark: The solution relies on the fact that you know the true value of : in reality, it is usually the case that such value is unknown. So in order to compute the sample size n to achieve a desired margin of error, the value of is either estimated via the sample standard deviation S, or it is very common to use the standard deviation from a previous study. Consequently, the n that you find is inevitably an approximation.