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. The correct computation of confidence intervals requires that the distribution of the observations be known. But very often approxi- mations are close enough to correctness to be of use, and often are, or may be made to be, con- servative. For computation of confidence inter- vals for the mean, the normal distribution is usually assumed to apply for several reasons: the central limit theorem assures us that with large samples the mean is likely to be approximately normally distributed; the required computations are well known and are easily applied; and when the normal distribution is known not to apply, suitable transformation of the data often is avail- able to allow a valid application. The confidence interval for a mean is an inter- val within which the true mean is said to have some stated probability of being found. If the probability of the mean not being in the interval is a (a could equal .1, .05, .01 or any probability value), then the statement may be written P (CLl < P.<CL2)= 1- Q This is read, "The probability that the lower con- fidence limit (CLl ) is less than the true mean (JJ.) and that the upper confidence limit (C~) is greater than the true mean, equals 1 - a." How- ever, we never know whether or not the true mean is actually included in the interval. So the confidence interval statement is really a state- ment about our procedure rather than about JJ.. It says that if we follow the procedure for re- peated experiments, a proportion of those ex- periments equal to a will, by chance alone, fail to include the true mean between our limits. For example, if a = .05, we can expect 5 of 100 confidence intervals to fail to include the tnie mean. To compute the limits, the sample mean, X; the standard error, sj{; and the degrees of freedom, n-l; must be known. A ta n- 1 value from tables of Student's t is obtained corre- sponding to n-l degrees of freedom and probability a. The computation is . CLl = X - (ta) (sj{) C~ =x + (ta) (sj{) Other confidence limits may be computed, and one additional confidence limit is given in BIOMETRICS - CONFIDENCE INTERVALS this section - the confidence limits for the true variance, a2. The information needed here is similar to that needed for the mean, namely, the estimated variance, S2; the degrees of freedom, n-l . and values from X2 tables. The values from , X 2 depend upon the degrees of freedom and upon the probability level, a. The confidence interval is P [(n-. 1)82 ~ a2 ~ (n-1)s2 J' = 1- Q 2 - - 2 X ~ X l-~ 2 2 This will be illustrated for a = .05; (n- 1) = 30; and S2 = 5. Since a = .05; 1 - ~ = 0.975; the associated X2 975 = 16.8 and the X20 025 = 47.25. Thus, the probability statement for the variance in this case is P ~.19:5 02 ~ 8.93] = .95 7.0 LINEAR REGRESSION AND CORRE- LA nON 7.1 Basic Concepts It is often desired to investigate relationships between variables, i.e., rate of change of biomass and concentration of some nutrient; mortality per unit of time and concentration of some toxic substance; chlorophyll and biomass; or growth rate and temperature. As biologists, we appreciate the incredible complexity of the real- world relationships between such variables, but, simultaneously, we may wish to investigate the desirability of approximating these relationships with a straight line. Such an approximation may prove invaluable if used judiciously within the limits of the conditions where the relation holds. It is important to recognize that no matter how well the straight line describes the data, a causal relationship between the variables is never implied. Causality is much more difficult to establish than mere description by a statistical relation. When studying the relationship between two variables, the data may be taken in one of two ways. One way is to measure two variables, e.g., measure dry weight biomass and an associated chlorophyll measurement. Where two variables 19