bstat exam 3

Central Limit Theorem for means

For n ≥ 30, sampling distribution of X̄ is approximately normal regardless of population distribution.

CLT for proportions requirement

np ≥ 5 and n(1-p) ≥ 5

Standard error of sample mean

σ/√n - measures how much sample means vary from population mean.

Standard error of sample proportion

√[p(1-p)/n] - measures how much sample proportions vary.

Sampling error

Natural random variation because we sample instead of surveying everyone.

Non-sampling error

Human mistakes, poor design, measurement problems, bias.

Selection bias

Systematic tendency for some groups to be over/under-represented in sample.

Nonresponse bias

Systematic difference between respondents and non-respondents.

Response bias

Systematic pattern of inaccurate answers (social desirability, leading questions).

Simple Random Sample (SRS)

Every possible sample of size n has equal chance of selection.

Stratified sampling

Divide population into homogeneous groups, then take SRS from each group.

Cluster sampling

Divide population into heterogeneous clusters, randomly select clusters, sample all within chosen clusters.

Unbiased estimator

E(θ̂) = θ - expected value equals population parameter.

Consistent estimator

Gets closer to true parameter as n → ∞ (both unbiased and variance → 0).

Relative efficiency

Among unbiased estimators, prefer one with smaller variance.

Z-test for means formula

z = (X̄ - μ₀)/(σ/√n).

Sample mean in z-test formula

X̄ in z-test formula.

Hypothesized population mean under null hypothesis

μ₀ in z-test formula.

Known population standard deviation

σ in z-test formula.

Sample size in z-test formula

n in z-test formula.

Square root of sample size

√n in denominator affects standard error.

T-test for means formula

t = (X̄ - μ₀)/(s/√n).

Sample standard deviation in t-test formula

s in t-test formula (used when population σ unknown).

Degrees of freedom in t-test

df in t-test = n - 1.

Z-test for proportions formula

z = (p̂ - p₀)/√[p₀(1-p₀)/n].

Alternative hypothesis (Hₐ)

Research hypothesis, effect exists, difference

Type I error (α)

Rejecting H₀ when it's actually true (false positive)

Type II error (β)

Failing to reject H₀ when it's actually false (false negative)

Power of test

1 - β = probability of correctly rejecting false null hypothesis

p-value definition

Probability of obtaining results as extreme as observed, assuming H₀ is true

Rejection rule

Reject H₀ when p-value < α (significance level)

Statistical significance

Result unlikely due to chance alone (p-value < α)

Practical significance

Result is large enough to be important in real world

One-tailed test

Tests for direction (Hₐ: μ > μ₀ or Hₐ: μ < μ₀)

Two-tailed test

Tests for difference (Hₐ: μ ≠ μ₀)

α = 0.05

Standard significance level - 5% chance of Type I error

α = 0.10

Less strict significance level

α = 0.01

More strict significance level

Z-test vs t-test choice

Z if σ known; t if σ unknown (almost always t-test in practice)

When normal assumption crucial

When n < 30 - for large samples, CLT applies regardless of population shape

Relationship: CI and two-tailed test

If CI contains μ₀ → fail to reject H₀; if excludes → reject H₀

One-tailed vs CI mismatch

CI always two-tailed, so may disagree with one-tailed test conclusion

Test statistic interpretation

How many standard errors sample result is from null value

Large test statistic

Evidence against H₀ (sample result far from null value)

Small p-value

Strong evidence against H₀ (result unlikely if H₀ true)

Large p-value

Weak evidence against H₀ (result likely even if H₀ true)

Fail to reject H₀ conclusion

"Evidence does not support" the alternative hypothesis

Reject H₀ conclusion

"Evidence supports" the alternative hypothesis