# Statistics for Environmental Engineers

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The t distribution is bell-shaped and symmetric (like the normal distribution), but the tails of the

t distribution are wider than tails of the normal distribution. The width of the t distribution depends on

2

the degree of uncertainty in s , which is measured by the degrees of freedom v on which this estimate of s2 is based. When the sample size is infinite (v = ^), there is no uncertainty in s2 (because s2 = a2and the t distribution becomes the standard normal distribution. When the sample size is small (v < 30), the variation in s2 increases. This is reflected by the spread of the t distribution increasing as the number of degrees of freedom of s2 decreases. The tail area under the bell-shaped curve of the t distribution is the probability of t exceeding a given value. A portion of the t table is reproduced in Table 2.2.

The conditions under which the quantity t = (y — n)/s has a t distribution with v degrees of freedom are:

1.    y is normally distributed about n with variance a2.

2.    s is distributed independently of the mean; that is, the variance of the sample does not increase or decrease as the mean increases or decreases.

3.    The quantity s2, which has v degrees of freedom, is calculated from normally and independently distributed observations having variance a .