Stats exam #2

Parameter

Number that describes the population. Its a fixed number and we dont know the value. EX( mu or stdev)

Statistic

A statistic is a number that describes a sample from a population. Unlike a parameter, its value can vary depending on the sample selected. Ex: (x or Xbar)

What happens to the sampling distribution as n increases

The sampling distribution becomes narrower and approaches a normal distribution according to the Central Limit Theorem.

Sampling variability

How a statistic (like the mean) changes from sample to sample.

Sampling distribution

The distribution of all possible sample means from repeated samples of size n.

Effect of sample size on sampling distribution

Larger n → less variability → narrower distribution.

Standard Error of the Mean (SEM)

Standard deviation of the sampling distribution of the mean.

Why does independence matter?

Ensures valid probability calculations and correct standard error.

When is independence violated?

When sampling without replacement from a small population.

Central limit theorem definition (CLT)

The sum or average of many small random quantities gives distribution that is close to normal

CLT requirement

A very large sampling size. typically, n>30

CLT implication

Allows use of normal probability methos even if the population is not normal

Statistical inference

Draws conclusions about a population based on sample data (such as estimates of the population mean μ from our sample mean x).

Effect of sample size on CI

Larger n → wider interval

effect of confidence level on CI

Higher confidence → wider interval

effect of standard deviation on CI

Larger σ → wider interval.

90% vs 95% vs 99% CI

90% = narrowest, 99% = widest.

How to reduce margin of error (3 ways)

1. Using a lower level of confidence (smaller C).
2. Increasing the sample size (larger n).
3. Reducing σ

If we know the population’s underlying standard deviation, will the SEM change or vary with an individual’s group experiment? How about the width of the confidence interval?

Width stays the same, the center of the CI will change

Null hypothesis (Ho)

Assumes no effect or no difference

Alternative hypothesis (Ha)

What you are trying to find evidence for

Correct Ho form …

Must include equality (=, ≤, ≥).

One sided test

Tests for direction (>, <)

Two-sided test

Tests for any difference (≠)

Significance test

A formal procedure for comparing observed data with a hypothesis whose truth we want to assess. The results of the test are expressed in terms of a probability that measures how well the data and the hypothesis agree.

When to use z-test

σ known and/or large n

When to use t-test

σ unknown and small n

More conservative test (z or t)

t-test (heavier tails)

P-value definition

Probability of observing result as extreme (or more) assuming H₀ is true

Reject H0 if

if the p-value < α and accept Ha

Small p-value means

Evidence against H₀

Decision rule

If p ≤ α → reject H₀

If p > α

Fail to reject H₀

Reject H₀

Evidence supports Hₐ

Fail to reject H₀

Not enough evidence (NOT proof H₀ is true)

Type I error (False positive)

If we reject H0 (accept Ha) when in fact H0 is true

Type II error (False Negative)

If we fail to reject H0 (withhold judgment) when in fact
Ha is true