Hypothesis Testing

sampling distributions of a mean (SDM)

describes the bahvior of a sampling mean

x~N(μ, SE_x) where SE_x=σ/root(n)

hypothesis testing

tests a claim about a parameter using evidence (data in a sample)

also called significance testing

the techniques introduced by considering a one-sample z test

hypothesis testing steps

1. null + alternative hypothesis

2. test statistic

3. p-value + interpretation

4. significance level

null + alternative hypothesis

null hypothesis (H₀) is a claim of ‘no difference in the population’

alternative hypothesis (H_a) claims H₀ is false

collect data + seek evidence against H₀ as a way of bolstering H_a (deduction)

test statistic

use z_stat where μ₀=population mean assuming H₀ is true + SE_x=σ/root(n)

p-value

probability of the observed test statistic or one more extreme when H₀ is true

corresponds to area under the curve in the tail of the standard normal distribution beyond the z_stat

the smallest α level at which H₀ can be rejected

convert z statistics to p-value

for H_a: μ>μ₀ → P=Pr(Z>z_stat) = right tail beyond z_stat

for H_a: μ<μ₀ → P=Pr(Z<z_stat) = left tail beyond z_stat

for H_a: μ/=μ₀ → P=Z* one-tailed p-value

p-value interpretation

smaller -value = stronger evidence against H

α level

let α = probability of erroneously rejecting H_o

set α threshhold

reject H₀ when P <= x

retain H₀ when P > x

z test

σ known (not from data)

population approximately normal or large sample (central limit theorem)

simple random sampling

data vlid

motivation for statistical hypothesis testing

we want to test is a result is simply due to chance or deeper effect

hypothesis testing format

state hypothesis

use data to attempt to disprove the null hypothesis

draw conclusions based on result (often using the p-value) → always state conclusions with respect to null hypothesis

two-sided test

the null hypothesis allows any value of a parameter larger (or smaller) than a specifed value

one-sided test

the null hypothesis asserts a specific value for the population parameter

z test for proportion

x = μ + zσ