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<br />. <br /> <br />from a population distributed as a normal distri- <br />bution. Even if the population is not distributed <br />normally, however, as sample size increases, the <br />t-test approaches to applicability. If it is <br />suspected that the population deviates too <br />drastically from the normal, exercise care in the <br />use of the t-test. One method of checking <br />whether the data are normally distributed is to <br />plot the observations on normal probability <br />graph paper. If the plot approximates a straight <br />line, using the t-test is acceptable. <br /> <br />The t-test is used in certain cases where it is <br />known that the parent distribution is not <br />normal. One case commonly encountered in <br />field studies is the binomial. The binomial may <br />describe presence or absence, dead or alive, male <br />or female, etc. <br />Testing Ho : P = K (the population proportion <br />equals some value K): <br /> <br />p- K <br />t=- <br />-fi <br /> <br />(31 ) <br /> <br />where P = the symbol for the population propor- <br />tion (e.g., proportion of males in the popula- <br />tion); K = a constant positive fraction as the <br />hypothesized proportion; p = the proportion <br />observed in the sample; q = the complementary <br />proportion (e.g., the proportion of females in <br />the sample or 1 - p); and n = the number of <br />observations in the sample. Note that since p is <br />computed as (number of males in the sample) / <br />(total number of individuals in the sample), it <br />will always be a positive number less than one, <br />and hence, so will q. Again IX must be chosen; Ha <br />can be any of the types previously discussed; <br />and the degrees of freedom are n - 1. <br />Count data, where the objects counted are <br />distributed randomly, follow a Poisson distribu- <br />tion. If the Poisson can be used as an adequate <br />description of the distribution of the popula- <br />tion, an approximate t may be computed. <br />Testing Ho : Jl = M for the Poisson (the mean <br />of the population distributed as a Poisson equals <br />some hypothesized value M): <br /> <br />BIOMETRICS - CHI SQUARE TEST <br /> <br />Note that X = a2 for the Poisson, thus~S the <br />standard deviation of the mean, sj( . <br /> <br />5.2 Chi Square Test (x2-test) <br /> <br />Like t, X2 values may be found in mathe- <br />matical and statistical tables tabulated in a two- <br />way arrangement. Usually, as with t, the column <br />headings are probabilities of obtaining a larger <br />X 2 value when Ho is true, and the row headings <br />are degrees of freedom. If the calculated X2 ex- <br />ceeds the tabular value, then the null hypothesis <br />is rejected. The chi square test is often used with <br />the assumption of approximate normality in the <br />population. <br />Chi square appears in two forms that differ <br />not only in appearance, but that provide formats <br />for different applications. <br />. One form: <br /> <br />2 (n-l)s2 <br />X = ---or- <br /> <br />(33) <br /> <br />is useful in tests regarding hypotheses about a2 . <br />. The other form: <br /> <br />X2 = L (O-EE)2 (34) <br /> <br />where 0 = an observed value, and E = an ex- <br />pected (hypothesized) value, is especially useful <br />in sampling from binomial and multinomial <br />distribution, Le., where the data may be classi- <br />fied into two or more categories. <br />Consider first a binomial situation. Suppose <br />the data from fish collections from three lakes <br />are to be pooled and the hypothesis of an equal <br />sex ratio tested (Table 4). <br /> <br />TABLE 4. POOLED FISH SEX <br />DATA FROM 3 LAKES <br /> <br />No. males <br />892* (9l9)t <br /> <br />No. females <br />946 (919) <br /> <br />Total <br />1838 <br /> <br />*Observed values. <br />t Expected, or hypothesized, values. <br /> <br />. <br /> <br />X-M <br /> <br />t=-Jr <br /> <br />To compute the hypothesized values (919 <br />above), it is necessary to have formulated a null <br />(32) hypothesis. In this case, it was <br /> <br />"0 : No. males = No. females = (.5) (total) <br /> <br />13 <br />