stats midterm 2

(uniform) height formula

1/b-a

p(a1 < x < a2) (for uniform)

= (a2-a1)/ (b-a)

p(x<6) (for uniform)

(6-a)(b-a)

E(x) (for uniform)

(a+b)/2

Var(x) (for uniform)

(b-a)²/2

cdf problem

1. set integral = 1

2. plug probabilities into integral (cdf)

3. multiply integrals by x and again by x²

4. var(x)= x solution - x² solution

exponential distribution

e(X)= 1 / λ

r command to exp dist

pexp(x, rate= λ)

gamma distribution

o²/u= B

gamma r command

pgamma(p, shape= a, scale= b)

standardize and find p r command

pnorm(x, mean(u), sd(o))

unstandardize and find x r command

qnorm(p, mean(u), sd(o))

empirical rule 68

u-o u u+o

standard normal

0 and 1

find c

qnorm(1- (1- 0.9) /2)

sample distrubution

pnorm(x, mean, sd/square root of n)

one sample t interval

estimating a mean, and if o is unknown and only know s,

find t then plug into formula

one sample t interval r command

qt( (1- a/2) , df (n-1))

one sample z interval

estimating mean, o is known

find z then plug in to E= z ((o)/sqrt of n)

one sample z interval r command

qnorm(1-a/2)

to find E in z or t interval

E= (t or z) x (s or o)/ square root of n

reverse t interval problem

when you get the interval, but no x or e, or s

x= sum/2

e= difference/2

e= t (s/ sqrt n)

proportion z interval

want a proportion, but no p or n

e= width/2

if no p, use: 0.5

n= (z/e)² p(1-p)

𝝀² interval for variance

var(c(…..))

𝝀²= o²/2

left= qchisq( 𝝀², df)

right= qchisq( 1- 𝝀², df)

then plug into formula right, o², then left

one sample z test for means

testing for mean, but o is known

find z through formula

find pval, with pnorm(z), if two tailed: 2 times pnorm)-abs(z))

one sample t test for means

testing for mean, but σ is unknown (only s given).

find t through formula.

find pval with pt(t, df=n-1). df = n−1.

one proportion z test

testing a proportion (p), not a mean.

p̂ = x/n.

find z through formula — denominator uses p₀ not p̂.

find pval with pnorm(z).

two sample t test for means

comparing means from two independent groups.

find t through formula.

find pval with pt(t, df) — df comes from Welch's formula (messy, usually given or use R).

paired t test for means

comparing means but same subjects measured twice (before/after, computer/manual).

compute differences D first, then it's just a one sample t test on D.

find t with D̄ and s_D.

find pval with pt(t, df=n-1) where n = number of pairs.

two proportion z test

comparing proportions from two independent groups.

p̂₁ = x₁/n₁, p̂₂ = x₂/n₂.

find z through formula. find pval with pnorm(z).

note: the denominator uses each group's own p̂, not a pooled p.