stats exam 2 mc

What is a random sample?

A sample where each individual in the population has an equal chance of being selected.

Why is random sampling important?

It prevents bias and ensures the sample represents the population.

What two conditions must be met for a sample to be random?

Each selection must be independent

What is a common violation of random sampling?

Selecting participants based on convenience.

What is the shape of a normal distribution?

Symmetrical and bell-shaped

What percent of scores lie between the mean and ±1 SD?

68% total (34% on each side).

What percent lie between ±2 SD?

About 95%.

What percent lie between ±3 SD?

About 99.7%.

Example of SAT mean and SD?

μ = 500

What is the formula for probability?

P(A) = # of desired outcomes / total # of outcomes.

Probability of selecting a female from 12 females and 8 males?

12/20 = 0.6 (60%).

What range can probabilities fall between?

0 and 1.

How does increasing total outcomes affect probability?

Each single outcome becomes less probable.

What is the expected value of the sample mean (M)?

E(M) = μ (the population mean).

Why is the sample mean unbiased?

Because

What does the expected value represent?

The mean of all possible sample means.

What is the distribution of sample means?

The distribution of all possible sample means for samples of size n.

What does the Central Limit Theorem state?

For n ≥ 30

When is the distribution of sample means normal?

When the population is normal or n ≥ 30.

Why is this important?

It allows us to apply z-scores and probabilities to sample means.

What is the formula for standard error (σM)?

σM = σ / √n.

What does standard error measure?

The average distance between sample means and μ.

How does sample size affect σM?

Larger n → smaller σM → more accurate estimates.

How does population variance affect σM?

Larger σ → larger σM.

What law supports this concept?

The Law of Large Numbers.

Probability of guessing correctly on a 4-choice question?

1/4 = 0.25.

Expected correct out of 20 questions with 4 choices each?

20 × 0.25 = 5 correct.

What is the range of probabilities?

0 to 1.

What does the Law of Large Numbers state?

As sample size increases

Why is this important?

Larger samples are more representative and stable.

Formula for z using sample means?

z = (M − μ) / σM.

What does the z-table show?

Proportions and probabilities under the standard normal curve.

What is the probability above z = 2.00?

About 0.0228 (2.28%).

Why compute σM first?

Because it depends on sample size n.

What is a Type I error?

Rejecting a true null hypothesis (false positive).

What is a Type II error?

Failing to reject a false null hypothesis (false negative).

Symbol for Type I error probability?

α (alpha).

Symbol for Type II error probability?

β (beta).

Which type of error is more serious?

Type I error.

What are the 4 steps of hypothesis testing?

1. State H₀ and H₁ 2. Set α level 3. Compute test statistic 4. Decide to reject or fail to reject H₀.

What does the null hypothesis (H₀) state?

There is no effect or difference.

What does the alternative hypothesis (H₁) state?

There is a change or effect.

When do you reject H₀?

When z or t falls in the critical region (p < α).

What does “fail to reject H₀” mean?

There isn’t enough evidence for a difference.

What does “statistically significant” mean?

The result is unlikely to occur by chance (p < α).

Does significance mean importance?

No — large samples can make small effects significant.

How to report significance?

Example: z = 2.45

What does effect size measure?

The strength or magnitude of a treatment effect.

Formula for Cohen’s d (z-test)?

d = (M − μ)/σ.

Cohen’s effect size guidelines?

Small = 0.2

Formula for r² (t-test)?

r² = t² / (t² + df).

What does r² tell us?

The percent of variance explained by the treatment.

What does the alpha level (α) represent?

The probability of making a Type I error.

Common alpha levels?

.05

Lowering α does what?

Reduces Type I error risk but makes significance harder to reach.

What is the critical region?

The extreme area of scores where H₀ is rejected.

Difference between one-tailed and two-tailed tests?

One-tailed predicts direction; two-tailed checks for any difference.

When do we use a t-test instead of a z-test?

When σ (population SD) is unknown.

Formula for t statistic?

t = (M − μ)/sM

What are degrees of freedom (df)?

n − 1.

As df increases

the t distribution becomes what?

How does sample size affect power?

Larger n increases power (easier to detect effects).

Assumptions of a t test?

Independent observations and roughly normal population.

How does sample variance affect t?

Larger variance → smaller t → harder to find significance.

Confidence interval formula?

μ = M ± t sM.

What affects the width of a confidence interval?

Confidence level and sample size.

What does the Central Limit Theorem connect to?

Standard error and the normal shape of sample means.

What does a z-score of 0 mean?

The score equals the mean (μ).

What happens to standard error as n increases?

It decreases.

What is the range of probabilities?

0 to 1.

What does a large effect size indicate?

A strong and meaningful treatment effect.