Estimation and Hypothesis Testing

Size of the population

N

Mean of the population

µ

Variance of the population

σ²

Standard deviation of the population

σ

Proportion of the population

P

Standard Error

σ_s = σ/√n

Parameter notations

N µ σ² σ P

Statistic Notations

n x̄ s² s p̂

Size of sample

n

Mean of sample

x̄

Variance of sample

s²

Standard deviation of sample

s

Proportion in sample

p̂

Estimation

process of finding likely value for a population parameter based on a sample

Point Estimate

single valued estimate of a population parameter like mean or proportion

Interval Estimate

range of values for population parameter with a ± 95% CI

A better estimate is

narrow or tight estimate

lower variation or high sample size

A worse estimate is

wide or loose estimate

higher variation or less sample size

Standard Error or margin of error

estimation of the mean parameter in the population

Calculate confidence interval for large n>120

µ = x̄ ± Z_0.95 σ/√n

Calculate confidence interval for small n<120

µ = x̄ ± t_0.95 s/√n

Hypothesis Testing

process of identifying something of concern then generating a null and alternative hypothesis

Null hypothesis H₀

non-biased statement about value of a population parameter

assumed true until tested

Alternative Hypothesis H₀/H_A

what the researchers want to show

negates the H₀

One-tailed test

test H₀ in which H₁ only has one end

little sliver on graph 2.5%

One-tailed test Results

H₀ µ ≥ 12 and H₁ µ < 12

or

H₀ µ ≤ 12 and H₁ µ > 12

Two-tailed Test

test of H₀ in which H₁ has two ends

both slivers on the graph 5%

Two-tailed Test Results

H₀ µ =12 or H₀ µ ≠ 12

If P-value < 0.05

reject the H₀

If P-value≥ 0.05

fail to reject (FTR) H₀

If CI includes null or 0

fail to reject (FTR) H₀

If CI excludes null value or 0

reject the H₀

Type 1 or alpha error

likelihood of incorrectly reject H₀

false positive

Type 2 or beta error

likelihood of incorrectly FTR H₀

false negative

Minimizing Type 1 errors

controlling bias in study design and adjusting for cofounding

Minimizing Type 2 errors

increase the sample size to reject the null

One-Sample T-test or Parametric Test is the

compare the sample mean to an expected mean population

Use One sample or Parametric test if

the 1 scale variable is normally distributed

Reporting One Sample T-test

test value mean±SD P-value CI

Median or Runs Nonparametric test

used if scale variable is not normally disturbed

Reporting Runs Test

test value median (Q₁-Q₂) P-values