Cheatography

# Quantitative Methods Final Exam Cheat Sheet by rockcollector2

### Quartiles

 IQR = Q3 = Q1 Outliers are beyond: Q1 - 1.5 * IQR, Q3 + 1.5 + IQR

### Calc Entry for Basic Stats

 1-VAR Stats

### >CO­MBI­NAT­ORIAL PROBAB­ILI­TY<

 4-Digit PIN with repetition = 10^4 4-Digit PIN without repetition = 10!/6! = 5040

### Permut­ations

 How many ways can 6 people be ranked? Ranking n objects leads to n! How many ways can 6 people be ranked into 3 places: n(n-1)...*(n­-r+1) = n!/(n-r)! or 6 nPr 3

### Combin­ations

 Combin­ations = choosing a certain number of objects from a given set (no order). N choose R or N!/(N-­R)!R! or nCr

### Example

 3 cities from 15 are chosen randomly for a visit. A: 4 cost \$800, B: 5 cost \$300, C: 6 cost \$100. What is the probab­ility that the tour will cost \$1000 or less? - All from C 6 nCr 3 - Two from C, one from B or A (6nCr2­)(5­nCr1) + (6nCr2­)(4­nCr1) - One from C, two from B (6nCr1­)*(­(5nCr2) - None from C, three from B (5nCr3) Compute and sum: 20+75+­60+­60+­30=245; Divide by the total 15nCr3 = 455 .538 or 53.8%

### >RANDOM VARIAB­LES<

 Example of Random Variable X is the number of heads in 10 flips of a coin. P(X=4)? (10nCr­4)/2^10 = .205 X is sum of two dice. P(X=5) = 4/36 P(9<=X­<=11) = P(X=9)­+P(­X=1­0)+­P(X­=11)= 4/36+3­/36­+2/36

### Binomial Random Variables

 Binomial Random Variable represents the number of successes in n trials with probab­ility p of success. The probab­ility of 4 successes in 10 trials (p=0.5) is (10 nCr4_/2^10 = 0.205 Calc: binomp­df(­10,.5,8) = (n,p,r) r is number of successes Math: (n r) pr(1-p)n-r

### Binomial Random Variables

 Binomial with n=15 and p = .4 Compute: P(X=3)­=bi­nom­pdf­(15­,.4,3) P(X<=3­)=b­ino­mcd­f(1­5,.4,3) P(X<3)­=bi­nom­cdf­(15­,.4,2) P(X>3)­=1-­bin­omc­df(­15,.4,3) P(X>=3­)=1­-bi­nom­cdf­(15­,.4,2) P(4

### Continuous & Normal Random Variable

 For C RV, P(X=3) or any other value is zero. Probab­ility is area under curve. Normal RV is bell curve. Total area under bell is 1. P(X

### Pure Numbers

 X is a random normal variable with mean -3 and standard deviation 0.7 P(-4-2) = normal­cdf­(-2­,1E­99,­-3,.7) P(X<=-­3.5­)=n­orm­alc­df(­-1e­((,­-3.5­,-­3,.7) P(X=-3) =0 P(|X-(­-3)­|>.7­)=­P(X­<-3.7)­+P(­X>-2.3) = normal­cdf­(-1­E99­,-3.7,­-3,.7)­+no­rma­lcd­f(-­2.3­,1E­99,­-3,.7) A car model gets 24 mpg on the car sticker. The maker knows that this is normally distri­buted with a std dev of 3 mpg. What is the proportion of cars that get less than 20 mpg? P(X,20­)=n­orm­alc­df(­-1E­99,­20,­24,­3)=0.91

### Condit­ional Probab­ility

 P(A|B) is the probab­ility of A given that B happened. P(A|B) = P(Aint­ers­ectB) /P(B) If P(A|B) = P(A) then indepe­ndent. P(Aint­ers­ectB) = P(A)P(B) If mutually exclusive P(A|B)=0

### Central Limit Theorem

 As n becomes large, the sample mean will be distri­buted according to the normal distri­bution with parameters u and standard deviation - std dev/sqrt n *As n gets large, the spread in the sample mean distri­bution narrows. This means that the sample mean is more likely to be near the true mean.

### >IN­FER­ENTIAL STATIS­TIC­S<

 Z-test approx­imated by normal distri­bution. If sample size is large or variance is known. T-scor­e/test is used when: - sample size is below 30 - population standard deviation is unknown (estimated from your sample data) otherwise use z-scor­e/test. Generally use 95% confidence level. Z-Stat represents how many std. deviations away from the mean the sample mean is. Std. dev is std dev/sqrt n Null hypotheses assumes that whatever you are trying to prove did not happen. p-value of 0.03 means there is a 3% chance of finding a difference as large as or larger than the one in your study given the null hypothesis is true. If 0.05 or less you typically do not accept the null hypoth­esis. Type 1 error: rejecting the null hypothesis when true Type 2 error: accepting the null hypothesis when false

### Newton's Method

 1. Pick a, initial guess. 2. Compute tangent line approx­ima­tion: y = f(a)+f­'(a­)(x-a) 3. Solve y=0 and get x = (f'(a)­a-f­(a)­)/f'(a) 4. Use x for the next guess. Repeat.

### Max/Min Word Problems

 10 meters of string. maximum area dimens­ions? Perimeter: P(l,w) = 2l_2w P=10 Area: A9l,w)=lw a(l)=l­(5-1) max is at l=2.5

### Critical Points

 Min: f goes from decreasing to increasing f' goes from negative to positive. Max: f goes from increasing to decrea­sing. f' goes from positive to negative. Flat: f continues to change in the same way. f' does not change sign. f" gives concavity. Concave up means second derivative is positive which means first derivative is increasing Concave down: f"<0, f' decrea­sing.

### Limits

 A function f(x) converges to a limit L at x=a if, for any given error tolerance, we can specify a range of x such that for any x in that range, f(x) is near L, near being given the tolerance.

### Computing Areas & Integrals

 The definite integral of f from a to b is the area underneath the curve from a to b. Where f is negative, the area contri­buted is a negative area. Use fnInt

### Deriva­tives and Tangent Lines

 The derivative of a function f at x is the slope of the tangent at x. If all of the slopes are assembled you get f'(x) or df/dx. If we know f'(a) and f(a), the the tangent line at x=a is y=f'(a­)(x­-a)­+f(a) Slope is f'(a) and line passes through (a,f(a)) Approx­imate slope use: nDeriv

### Calculus Notes

Nice sheet. I just wanted to clarify if your standard deviation formula is really the variance. You would take the square root of variance to get sigma which is standard deviation.