AP®︎ Statistics: Unit 3 Flashcards

An estimator is a rule or statistic used to estimate a population parameter; an unbiased estimator has a sampling-distribution mean equal to the true parameter

An estimator that produces a single numerical value as its estimate

A population parameter is a numerical characteristic of a population that is fixed but usually unknown; a sample statistic is a numerical value computed from sample data and used to estimate a parameter

p̂ = X/n, where X is the number of successes in the sample

The probability distribution of p̂ over all possible random samples of the same size n

The sampling distribution of p̂ has mean μ_p̂ = p and standard deviation σ_p̂ = √(p(1 − p)/n) when sampled observations are independent

Data should be collected using a random sample from the population

When sampling without replacement, N ≥ 10n (equivalently, n ≤ 0.10N) so sampled observations can be treated as approximately independent

Approximate normality for p̂ requires np ≥ 10 and n(1 − p) ≥ 10

The confidence interval procedure for estimating one population proportion p: p̂ ± z*√(p̂(1−p̂)/n)

An interval estimate for a population parameter; a range of plausible values for the parameter based on sample data

A statement that names the proportion, the success/response category, and the population of interest

The standard normal value with −z* and z* enclosing the middle C% of the distribution for the chosen confidence level

SE = √(p̂(1−p̂)/n); the estimated standard deviation of the sampling distribution of p̂, describing the typical sampling variability of p̂ around p

MOE = z*√(p̂(1−p̂)/n); the amount added to and subtracted from p̂, equal to half the interval width

Use n = (z*^2 p̂(1−p̂)) / MOE^2 and round up to the next whole number; if no reasonable estimate of p̂ is available, use p̂ = 0.5 for a conservative sample size

We are C% confident that the interval from L to U contains the true proportion of the specified population that has the specified characteristic

If many random samples of the same size were taken from the population and a C% confidence interval were calculated each time by the same method, about C% of those intervals would contain the true population proportion

For the same sample, higher confidence level → larger z* → larger margin of error → wider confidence interval

For a fixed confidence level and roughly the same p̂, larger n → smaller standard error → smaller margin of error → narrower confidence interval

Hypothesis test for one population proportion p using z = (p̂ - p₀)/√(p₀(1 - p₀)/n) when conditions are met

The null hypothesis is the statement about the population proportion assumed true unless the sample gives convincing evidence otherwise; the alternative hypothesis describes the departure from the null that the evidence is meant to support

A two-sided alternative is Hₐ: p ≠ p₀; a one-sided alternative is Hₐ: p < p₀ or Hₐ: p > p₀ depending on the research question

Randomization; 10% condition if sampling without replacement; normality under H₀: np₀ ≥ 10 and n(1 - p₀) ≥ 10

Distribution of the test statistic assuming H₀ is true; approximately standard normal when conditions are met

Probability, assuming H₀ is true, of getting a test statistic as extreme as or more extreme than the observed value in the direction of Hₐ.

The p-value is the probability, assuming the null hypothesis about the population proportion is true, of getting a sample result or test statistic at least as extreme as the one observed in the direction of the alternative

z = (p̂ - p₀)/√(p₀(1 - p₀)/n); measures how many null standard deviations p̂ is from p₀.

α, the predetermined probability of rejecting H₀ when H₀ is true

A result is statistically significant at level α when p-value ≤ α.

Reject H₀ means conclude there is convincing statistical evidence to support Hₐ; fail to reject H₀ means conclude there is not convincing statistical evidence to support Hₐ and does not prove H₀ true

State the conclusion in context about the population proportion, consistent with Hₐ, using non-definitive language such as 'there is convincing statistical evidence' or 'there is not convincing statistical evidence.'

A Type I error is rejecting a true H₀; a Type II error is failing to reject a false H₀

Probability of correctly rejecting H₀ when H₀ is false; power = 1 - β.

α is the probability of a Type I error, β is the probability of a Type II error for a specified true parameter value, and power = 1 − β; with other features fixed, decreasing α lowers Type I error but increases β and lowers power, while increasing α does the opposite

Holding other factors constant: larger sample size, smaller standard error, a true parameter value farther from the null value, or a larger α.

Distribution of the values of p̂1 − p̂2 over repeated independent random samples of the same sizes, or repeated random assignment in an experiment

The sampling distribution of p̂1 − p̂2 has mean μ_p̂1−p̂2 = p1 − p2 and standard deviation σ_p̂1−p̂2 = sqrt[p1(1 − p1)/n1 + p2(1 − p2)/n2]

Check randomization, the 10% condition when sampling without replacement in each population, and the large-counts condition that all four expected counts are at least 10

Confidence interval procedure for estimating p1 - p2 from two independent groups for the same binary response

Point estimate for p1 - p2; the observed difference between the two sample proportions

SE = sqrt(p̂1(1-p̂1)/n1 + p̂2(1-p̂2)/n2)

MOE = z*sqrt(p̂1(1-p̂1)/n1 + p̂2(1-p̂2)/n2)

(p̂1 - p̂2) ± z*sqrt(p̂1(1-p̂1)/n1 + p̂2(1-p̂2)/n2)

We are C% confident that the interval from a to b contains the true difference p₁ − p₂, where p₁ and p₂ are the population proportions for the two groups in context

If the interval for p1 − p2 is entirely above 0, there is convincing evidence that p1 > p2; if entirely below 0, there is convincing evidence that p1 < p2; if it contains 0, there is insufficient evidence that the population proportions differ

Significance test for whether an observed difference p̂1 − p̂2 provides convincing evidence of a difference between two population proportions

H₀: p₁ = p₂, equivalently p₁ − p₂ = 0; Hₐ: p₁ > p₂, p₁ < p₂, or p₁ ≠ p₂ depending on the question

Estimate of the common population proportion under H₀: p̂c = (x1 + x2)/(n1 + n2) = (n1p̂1 + n2p̂2)/(n1 + n2)

Check that data come from two independent random samples or a randomized experiment, each sample satisfies the 10% condition when sampling without replacement, and all four expected counts using p̂c are at least 10

z = (p̂1 − p̂2 − 0)/√[p̂c(1 − p̂c)(1/n1 + 1/n2)]; measures how many null standard errors the observed difference is from 0

Compare p-value with α, state reject H₀ or fail to reject H₀, and conclude in context in terms of Hₐ about the population proportions using non-definitive language

A family of right-skewed distributions with only nonnegative values; skew decreases as degrees of freedom increase

A chi-square test for homogeneity compares the distribution of one categorical variable across two or more populations or treatment groups, while a chi-square test for independence tests whether two categorical variables are associated within one population using one random sample

H₀: the distribution of the categorical variable is the same across populations or treatments; Hₐ: the distributions are not all the same

H₀: the two categorical variables are independent, or there is no association, in the population; Hₐ: they are not independent, or there is an association

Check randomization, the 10% condition when sampling without replacement, every expected cell count greater than 5, and that the table contains counts rather than percentages or measurements

Count expected in a cell if H₀ is true; expected count = (row total × column total) / table total; expected counts need not be whole numbers

χ² = Σ (O − E)² / E over all non-total cells; measures total disagreement between observed and expected counts

Df = (r − 1)(c − 1), using the number of non-total rows and columns

Probability, assuming H₀ is true in context, of getting a chi-square statistic at least as large as the observed value; a chi-square test for homogeneity or independence is always right-tailed

For independence: reject H₀ means convincing evidence of an association between the two categorical variables, while fail to reject H₀ means not convincing evidence of an association; for homogeneity: reject H₀ means convincing evidence that the distributions differ across populations or treatments, while fail to reject H₀ means not convincing evidence that they differ