When integrating by parts with one factor as a variable and the other with a trig function, generally the variable will be set to u and the trig function set to dv

integral udv =  uv - integral vdu

let's take integral x^2*sinx

u = x^2
du = 2x dx
dv = sinx
v = -cosx

this gives us  -x^2(cosx) - integral -2xsinx

now we have to integral by parts again

u = -2x
du = -2 dx
dv = sinx
v = -cosx


2xcosx - integral 2cosx = 2xcosx -(1/2)sinx

answer is -x^2(cosx) - 2xcosx + (1/2)sinx + C

When estimating the area under a curve using the left endpoints and right endpoints, if the function is increasing the left endpoint approximation underestimates the area, while the right endpoint estimation overestimates. This is vice versa if the function is decreasing. The most accurate of the estimates is the midpoint estimation.

Suppose you know that integral (0 to 3) f(x) dx = 6 , what is the integral (0 to 3)(f(x) + 5x)?

We can break down down into the sum of integrals

integral (0 to 3)f(x) dx + integral (0 to 3) 5x dx

= 6 + [(5/2)x^2) evaluated at 3... = 6 + (5/2)(9) = 6 + 45/2 = 28.5

Remember that removing an outlier can greatly affect the correlation between two variables.

Suppose x,y are as follows

x  1  1 1  2 2  2  3 3  3  1
y  8 9 10 8 9 10 8 9 10 10

This has a relatively strong negative correlation, but if you remove the data point (1,10), all you have left is a block of data with correlation coefficient of 0.

When finding correlation coefficient r by hand, as well as regression slope and intercept, you will see many calculations involving terminology you are unfamiliar with.

This is what is all means

sum(x) = all x's added together
sum(y) = all y's added together
sum(x^2) = square each x, then sum the total
sum(y^2) = square each y, then sum the total
sum(xy) = muitiply each xy value , then sum the total
(sumx)^2 = sum of the x's then square
(sumy)^2 = sum of the y's, then square
r^2 is just taking r and squaring it
y-bar is the average of the y's, which is sum(y)/n
x-bar is the average of the x's, which is sum(x)/n

Hypothesis test procedure

1. state the Ho and Ha
2. find test statistic z or t, for proportions we can use z if np > 5 and n(1-p) or nq > 5
3. find critical value for z or t using the appropriate chart
4. make the decision on the test, if using p-value, if p-value > significance level then do not reject. If p-value < significance level of the test, reject.
5. state the conclusion
Type I error is when you reject Ho when you should not reject Ho
Type II error is when you do not reject Ho when you should reject Ho
The power of the test is the probability of not committing a type two error, which is basically the probability of making the correct decision of rejecting Ho.