math final

plurality voting

every voter gets one vote and the candidate with the most votes wins

majority voting

candidate with a majority of first place votes wins have to have more than 50%

instant run off voting

fill in first second and third choice, candidate with the fewest first place votes is eliminated, and the process continues

borda count

• In this method, points are awarded to candidates based on their ranking: 1 point for last place, 2 points for second to last, etc.

• The candidate with the largest point total wins

fairness criteria

statements that should be true in a fair election

hamilton’s method

1. Determine how many people each representative should represent

   1. To do this we divide the total population of all the states by the total number of reps (this is called the divisor)

2. Divide each states population by the divisor to determine for many reps it should have (these are called quotas

3. Cut off decimal parts of all quotas (these are called lower quotas)

   1. Add up the remaining whole numbers. This will always be less of equal to the total number of reps

   2. Usually off by a few

4. Assuming that the total for lower quota was a bit smaller, we look at the decima, assign the remaining reps, one each, to the states whose decimal was the highest

jefferson’s method

1. Determine how many people each rep should rep

   1. Divisor: total number divided by states

2. Divide each county population by divisor to get quota

3. Cover decimal- lower quota

4. If the number of reps does not match, reduce the divisor and recalculate the quota allocation. continue doing this until the total in step 3 is equal to the total number of representatives. The divisor is the adjusted divisor now.

linear growth, common difference, and formulas

linear growth is growth by the same amount each time

common difference is the amt the population changes for each time period

formulas are:

recursive:pn=pn-1+d

explicict= pn=p0+dn

slope

y=mx+b

(m is the slope,b is the y-int)

y-intercept

b

explanatory and response variables

the explanatory variable (also called the independent variable) is used to explain or predict changes in another variable, the response variable (also called the dependent variable). Explanatory variables are typically manipulated by researchers in experiments, while response variables are the outcomes being measured

exponential growth

growth whose rate becomes ever more rapid in proportion to the growing total number or size.

formula:

	=	exponential growth function
	=	initial amount
	=	growth rate
	=	number of time intervals

simple interest

I=PxRxT

p=principle(starting amt)

r= annual percentage rate

t=time in years

compound interest

A=p(1+r/k)^kt

continuous growth

Pt=P0e^kt

savings annuity

The savings annuity formula calculates the future value of an investment where regular deposits are made over a set period, earning compound interest. The formula is: A = d[(1 + r/n)^(nt) - / (r/n). 

Here's a breakdown of the variables:

• A: The future value (or balance) of the annuity after t years.

• d: The regular deposit amount (or payment) made each period.

• r: The annual interest rate (as a decimal).

• n: The number of compounding periods per year.

• t: The number of years. 

payout annuity

The formula for a payout annuity calculates the present value (P) needed to fund a series of regular withdrawals (d) over a certain period (t), given an interest rate (r) and compounding frequency (n). The formula is: P = d * [1 - (1 + r/n)^(-nt)] / (r/n). 

Here's a breakdown of the variables:

• P: The initial principal or lump sum needed to start the annuity. 

• d: The regular withdrawal amount or payment. 

• r: The annual interest rate (as a decimal). 

• n: The number of compounding periods per year (e.g., 12 for monthly, 4 for quarterly, etc.). 

• t: The number of years the withdrawals will continue. 

loans

same formula as payout annuity

independent events

Imagine flipping a coin twice. The outcome of the first flip (heads or tails) doesn't change the probability of getting heads or tails on the second flip. These are independent events. 

Dependent events

Consider drawing two cards from a deck without replacement. The probability of drawing a specific card on the second draw depends on the outcome of the first draw. If the first card was a king, the probability of drawing another king on the second draw changes because there's one fewer king in the deck. 

sensitivity

This is the proportion of individuals with the disease who are correctly identified as having the disease by the test. A high sensitivity means the test is good at detecting the presence of the disease.

specificity

This is the proportion of individuals without the disease who are correctly identified as not having the disease by the test. A high specificity means the test is good at ruling out the presence of the disease. 

permutations, combinations, and neither

numerical date

Numerical data represents quantities that can be measured or counted.

categorical data

Categorical data describes characteristics that are not quantifiable and are typically represented by labels or categories. 

ordinal data

Ordinal data is categorical data where the categories have a meaningful order or ranking, but the intervals between categories may not be equal. 

nominal data

Nominal data is categorical data where the categories are simply labels or names with no inherent order or ranking. 

regressions, R, R²

In regression analysis, "R" and "R²" are statistical measures used to assess how well a regression model fits the data and how well it explains the variance in the dependent variable. R represents the correlation coefficient which indicates the strength and direction of the relationship between variables, while R² represents the coefficient of determination which indicates the proportion of variance in the dependent variable that can be explained by the independent variable(s). 

Standard Error for means

 (SD/sqrt(n))

confidence interval

mean+/- 2SE

Standard Error for proportions

(Sqrt(p(1-p)/n)

confidence interval

p+/- 2SE