1 Inferential Statistics http://www.slideshare.net/amenning/documents

Inferential Statistics: Confidence Intervals and Hypothesis Testing

1. Summary of Symbols and Definitions

Symbol Definition Symbol Definition

μ (mu) Population mean x ̄(x-bar) Sample mean

σ2 (sigma square) Population variance s2 Sample variance

σ (sigma) Population standard deviation s (also SD) Sample standard deviation

n Sample size df Degrees of freedom

σx ̄

(sigma x-bar) Standard deviation of the population of sample means

SE (for pop. mean)

Standard error: SE=SD/√n SE is our best to estimate of σx ̄

p Population proportion p ̂(p-hat) Sample proportion

(1-α)100% Confidence level α (alpha) Significance level

z Test statistic based on normal distribution

t Test statistic based on t-distribution

zα Normal point with right-hand tail area α : P(z ≤ zα) = 1-α

tα

tα, df

t-point with right-hand tail area α

(… and degrees of freedom df)

H0 Null hypothesis Ha Alternative hypothesis

μ0 Hypothesized value of the population mean

p0 Hypothesized value of the population proportion

2. Confidence Intervals for the Population Mean

All problems are based on a sample of size n randomly drawn from a population. The population mean μ

is unknown. The sample mean x̄ is used to estimate the population mean and the sample standard

deviation s is used to estimate the population standard deviation σ (in rare cases, σ may be known from

prior experiments). It is conventional to refer to the (exact or estimated) standard deviation as SD.

In all problems, the first step will be to calculate the standard error SE=SD/√n, which estimates the

standard deviation of the sampling distribution and is a measure of the accuracy of the sample mean. A

confidence Interval is always centered on the sample mean; its margin of error is a multiple of the

standard error.

Large Sample (n ≥ 30) Confidence Interval for population Mean:

CI at 95%: [x ̄ ± 𝟏. 𝟗𝟔 𝑺𝑬] = [x ̄ ± 𝟏. 𝟗𝟔 𝑺𝑫/√𝒏]

CI at 99%: [x ̄ ± 𝟐. 𝟓𝟕𝟓 𝑺𝑬] = [x ̄ ± 𝟐. 𝟓𝟕𝟓 𝑺𝑫/√𝒏]

CI at (1-α)100%: [x ̄ ± zα/2 𝑺𝑬] = [x ̄ ± zα/2 𝑺𝑫/√𝒏]

Small Sample CI: use the t-point tα/2 instead of the normal point zα/2 except if σ is known. The population

must be mound-shaped (approximately normal).

2 Inferential Statistics http://www.slideshare.net/amenning/documents

3. Confidence Interval and Hypothesis Testing for Sample Mean and Proportion

Sample Mean Sample Proportion

Basics

Estimate x̄ p ̂

Conditions (valid if) n ≥ 30 or population distribution normal (mound-shaped)

np ̂≥ 5 and n(1-p̂) ≥ 5

Standard error 𝑺𝑬 = 𝑺𝑫/√𝒏

Substitute SD = σ if known, otherwise SD = s

𝑺𝑬 = √𝒑(𝟏 − 𝒑)/𝒏

For CI substitute p = p̂ For HT substitute p = p0

Degrees of freedom Df = n-1 (for small sample)

Confidence interval (CI)

Confidence Level (1-α)100% (1-α)100%

Margin of Error E = tα/2, df SE or E = zα/2 SE if σ known or n large

E = zα/2 SE

CI at (1-α)100% CI = [x ̄± E] CI = [p ̂± E]

Hypothesis Testing (HT)

Significance level α α

Null Hypothesis H0 : μ = μ0 or μ - μ0 = 0 H0 : p = p0 or p - p0 = 0

Alternative Hypothesis Ha : μ ≠ μ0 (two-sided)

Ha : μ < μ0 (one-sided) Ha : μ > μ0 (one-sided)

Ha : p ≠ p0 (two-sided)

Ha : p < p0 (one-sided) Ha : p > p0 (one-sided)

Critical value tα/2, df (two-sided)

tα, df (one-sided)

Use z-point for large sample or if σ known

zα/2 (two-sided)

zα (one-sided)

Effect size x ̄- μ0 p ̂ - p0

Statistically significant effect / deviation

tα/2, df SE or zα/2 SE (two-sided)

tα, df SE or zα SE (one-sided)

zα/2 SE (two-sided)

zα SE (one-sided)

Test statistic t = (x ̄ - μ0 )/SE z = (p ̂- p0)/SE

Hypothesis test H0 is rejected if the effect size, i.e. the deviation of the parameter estimate from the hypothesized parameter, is statistically significant at significance level α. Otherwise H0 is not rejected.

Equivalently: H0 is rejected if the absolute value of the test statistic exceeds the critical value. For one-sided test, first check the sign of the deviation; use common sense!