exam 2 flash cards

Random Sample

A random sample means everyone in the population has the same chance to be picked.

Sampling Error

The small difference between a sample’s result and the whole population’s result that happens naturally.

Distribution of Sample Means

All the possible averages (means) you could get if you took many random samples from the same population.

Central Limit Theorem

When your sample size is big (usually 30+)

Standard Error (σₘ)

Shows how much the sample mean usually differs from the real population mean; σₘ = σ / √n.

Law of Large Numbers

The bigger the sample the closer the sample mean will be to the real population mean.

z-Score for Sample Means

Shows how far a sample mean (M) is from the population mean (μ) using z = (M – μ) / σₘ.

Using z-Scores to Find Probability

Use the unit normal table after finding z to see how likely that sample mean is.

Why Large Samples Are Better

Bigger samples make the standard error smaller so results are more accurate.

Hypothesis Testing

A method used to test if something really changes a population or if results happened by chance.

Goal of Hypothesis Testing

To decide if a sample’s result is strong enough to show a real effect in the population.

Steps of a z-Test

1 State H₀ and H₁; 2 Choose α; 3 Find z = (M – μ)/σₘ; 4 Decide to reject or not reject H₀.

Alpha Level (α)

The cut-off for what’s rare; example α = .05 means 5% risk of being wrong if we reject H₀.

Critical Region

The area of very unlikely results if H₀ is true; results there lead to rejecting H₀.

Type I Error

Rejecting a true H₀; a false positive.

Type II Error

Failing to reject a false H₀; a false negative.

Statistical Significance

Means the result is rare if H₀ is true so we reject H₀.

Effect Size (Cohen’s d)

Shows how big the difference is; d = (M – μ)/σ where .2=small .5=medium .8=large.

Statistical Power

The chance your test correctly finds a real effect; Power = 1 – β and increases with bigger samples.

Assumptions for z-Test

Need random samples independent scores normal distribution and known σ.

Why Use the t-Test

We use it when we don’t know the population standard deviation σ.

t-Test Formula

t = (M – μ) / sₘ where sₘ = s / √n.

Degrees of Freedom (df)

df = n – 1 shows how many scores can vary and affects the t distribution’s shape.

t-Distribution

Looks like a normal curve but with wider tails; becomes more normal with higher df.

Steps of a t-Test

1 State H₀ and H₁ 2 Choose α and find critical t 3 Compute s² and sₘ 4 Find t 5 Compare t and decide.

Assumptions for t-Test

Observations must be independent and the population should be roughly normal.

Sample Size and Variance Effect

Larger n makes error smaller and t bigger; larger variance makes error bigger and t smaller.

Effect Size for t-Test

Cohen’s d = (M – μ)/s or r² = t²/(t² + df) showing how strong the treatment effect is.

Confidence Interval

A range that probably contains the true mean; μ = M ± t × sₘ.

Directional (One-Tailed) t-Test

Used when predicting one direction such as an increase; only one tail has the critical region.

Chapter 7 Summary

Big samples make more accurate means and use z-scores to find probabilities.

Chapter 8 Summary

Hypothesis testing checks if results are real or random chance.

Chapter 9 Summary

t-Tests are used when σ is unknown and rely on sample information.