tom.h.wilson Dept. Geology and Geography West Virginia University

Making the t-test using tables of the critical values of t for various levels of confidence. But first - some review

The test statistic We have the means of two samples & is the standard error, But its computed differently from the single sample standard error which is just is the unbiased estimate of the standard deviation Note: How do we compute the critical value of t - the test statistic?

In this case where s p is the pooled estimate of the standard deviation derived as follows Similar form Going through this computation for the samples of strike at locations A and B yields t ~ 5.2.

Evaluating the statistical significance of differences in strike at locations A and B using the t-test

The value of our test statistic, t = 5.2 Degrees of freedom = n 1 +n 2 -2 = 38 Closest value in the table is 40

 = 0.1 % =0.001 as a one-tailed probability = 1 chance in 1000 that these two means are actually equal or were drawn from the same parent population.

PsiPlot returns a two- tailed probability for a t  5.2. That probability is (about one chance in 140,000). Note that this is twice the value returned by Excel.

The book works through the differentiation of y = x 2, so let’s try y =x 4. multiplying that out -- you get...

Remember this idea of dy and dx is that the differential changes are infinitesimal - very small. So if dx is (that’s 1x10 -4 ) then (dx) 2 = (or 1x10 -8 ) (dx) 3 = 1x and (dx) 4 = 1x So even though dx is infinitesimally small, (dx) 2 is orders of magnitude smaller

so that we can just ignore all those terms with (dx) n where n is greater than 1. Our equation gets simple fast Also, since y =x 4, we have and then -

Divide both sides of this equation by dx to get This is just another illustration of what you already know as the power rule,

is Just as a footnote, remember that the constant factors in an expression carry through the differentiation. This is obvious when we consider the derivative - which - in general for

Examining the effects of differential increments in y and x we get the following

Don’t let negative exponents fool you. If n is -1, for example, we still have or just

Given the function - what is? We just differentiate f and g individually and take their sum, so that

Take the simple example - what is? What are the individual derivatives of and?

let then - We just apply the power rule and obtain We know from the forgoing note that the c disappears.

We use the power rule again to evaluate the second term, letting g = (ax 4 +b) Thus -

Differences are treated just like sums so that is just

Recall how to handle derivatives of functions like ? or

Removing explicit reference to the independent variable x, we have Going back to first principles, we have Evaluating this yields Since df x dg is very small and since y=fg, the above becomes -

Which is a general statement of the rule used to evaluate the derivative of a product of functions The quotient rule is just a variant of the product rule, which is used to differentiate functions like

The quotient rule states that And in most texts the proof of this relationship is a rather tedious one. The quotient rule is easily demonstrated however, by rewriting the quotient as a product and applying the product rule. Consider

We could let h=g -1 and then rewrite y as Its derivative using the product rule is just dh = -g -2 dg and substitution yields

Multiply the first term in the sum by g/g (i.e. 1) to get > Which reduces to i.e. the quotient rule

The derivative of an exponential functions Given > In general for If express a as e n so that then Note

Sinceand in general a can be thought of as a general base. It could be 10 or 2, etc.

The derivative of logarithmic functions Given > We’ll talk more about these special cases after we talk about the chain rule.

Differentiating functions of functions - Given a functionwe consider writecompute Then computeand take the product of the two, yielding

We can also think of the application of the chain rule especially when powers are involved as working form the outside to inside of a function

Where Derivative of the quantity squared viewed from the outside. Again use power rule to differentiate the inside term(s)

Using a trig function such as let then Which reduces toor just

In general if then

Returning to those exponential and natural log cases - we already implemented the chain rule when differentiating h in this case would be ax and, from the chain rule, becomesor and finally since and

For functions like we follow the same procedure. Letand then From the chain rule we have hence

Thus for that porosity depth relationship we were working with -

For logarithmic functions like We combine two rules, the special rule for natural logs and the chain rule. Let Chain rule Log rule then and so

For next Tuesday answer question 8.8 in Waltham (see page 148). Find the derivatives of