AP®︎ Statistics: Unit 4 Flashcards

The probability distribution of x̄ for all possible random samples of a fixed size n from the same population; a distribution of sample means, not individual observations

For random sampling, the sampling distribution of x̄ has mean μx̄ = μ and standard deviation σx̄ = σ/√n when sampled values are independent

Data should be collected using a random sample so probability methods can describe how x̄ varies from sample to sample

When sampling without replacement from a finite population, require n/N ≤ 0.10, equivalently N ≥ 10n, so observations are approximately independent

If the population distribution is normal, then the sampling distribution of x̄ is normal for any sample size

If the population is not normal, the sampling distribution of x̄ becomes approximately normal as n increases; for this course, n ≥ 30 is usually enough unless the population is extremely skewed

Family of symmetric, bell-shaped, standardized distributions centered at 0 with lower peaks and heavier tails than the standard normal; identified by degrees of freedom

Parameter that identifies a t-distribution; for a one-sample t-procedure, df = n − 1

Confidence interval procedure for estimating μ from one sample of quantitative data when σ is unknown

Confidence interval procedure for estimating μ_d, the population mean of paired differences, by applying a one-sample t-interval to one sample of within-pair differences

The population mean of the quantitative response variable, stated in context

The population mean of the paired differences, stated in context with the subtraction order defined

Use a one-sample t-interval when data come from a random sample or randomized experiment, the 10% condition is met when sampling without replacement, and the population is approximately normal, or n ≥ 30, or if n < 30 the sample data show no strong skewness or outliers

A statistic used to estimate a population parameter; for a population mean it is x̄

SE = s/√n; estimated sample-to-sample variability of x̄

The half-width of a confidence interval; for a one-sample or matched-pairs t-interval, t*·s/√n

x̄ ± t*(s/√n), with df = n − 1

“We are C% confident that the interval from a to b contains the true population mean or true population mean difference for the population of interest,” stated in context; for paired data, include the order of subtraction

In repeated random sampling from the same population with the same sample size, about C% of confidence intervals constructed by this method will contain the true population mean or true population mean difference

If the claimed value is inside the confidence interval, the data are consistent with that value; if it is outside the interval, the interval gives evidence against that value

t* increases, so margin of error increases and the interval gets wider; the midpoint stays the same

Holding confidence level and variability approximately constant, increasing n decreases the standard error, so the margin of error and interval width tend to decrease

Significance test for a claim about a population mean μ when one quantitative sample is used and σ is unknown

One-sample t-test on the pairwise differences to test a claim about μd, the population mean difference, in a matched-pairs design

H₀: μ = μ₀; Hₐ: μ < μ₀, μ > μ₀, or μ ≠ μ₀, chosen from the investigative question

H₀: μd = 0; Hₐ: μd < 0, μd > 0, or μd ≠ 0, based on the context and defined differences

Use a one-sample t-test when data come from a random sample or randomized experiment, the 10% condition is met when sampling without replacement, and the population is approximately normal, or n ≥ 30, or if n < 30 the sample data are free of strong skewness and outliers

t = (x̄ − μ₀)/(s/√n)

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

If p-value ≤ α, reject H₀; if p-value > α, fail to reject H₀.

State the decision and a non-definitive conclusion consistent with H_a, naming the population, response variable, and population mean or mean paired difference; use wording like “there is convincing evidence” or “there is not convincing evidence,” not “proved” or “definitely is.”

The distribution of x̄1 − x̄2 from all possible independent random samples of sizes n1 and n2

For independent samples, the sampling distribution of x̄1 − x̄2 has mean μx̄1−x̄2 = μ1 − μ2 and standard deviation σx̄1−x̄2 = √(σ1²/n1 + σ2²/n2)

For x̄1 − x̄2, use two independent random samples or randomly assigned independent groups; if sampling without replacement, each sample must be no more than 10% of its population; the sampling distribution is normal if both populations are normal or approximately normal if n1 ≥ 30 and n2 ≥ 30

Confidence interval procedure for estimating μ1 − μ2 from two independent groups with a quantitative response when population standard deviations are unknown

μ1 − μ2: the difference between the two population means, stated in context with the response variable, populations, and subtraction order

Use a two-sample t-interval with two independent random samples or a randomized experiment with independent groups, with each sample no more than 10% of its population when sampling without replacement, and with an approximately normal sampling distribution of x̄1 − x̄2

SE = √(s1^2/n1 + s2^2/n2), the estimated variability of the difference in sample means

(x̄1 − x̄2) ± t*√(s1^2/n1 + s2^2/n2)

For an interval estimating μ1 − μ2, values inside the interval are plausible and values outside are not; if 0 is in the interval there is insufficient evidence the population means differ, and if all values are positive or all are negative, the sign shows which mean is larger

Test for a difference between two population means using a quantitative response from two independent groups when population standard deviations are unknown

State H₀: μ1 − μ2 = 0; the alternative is Hₐ: μ1 − μ2 ≠ 0, μ1 − μ2 > 0, or μ1 − μ2 < 0, depending on the question

Use a two-sample t-test with two independent random samples or a randomized experiment with random assignment, with each sample no more than 10% of its population when sampling without replacement, and with both populations approximately normal or both sample sizes at least 30 or, if either sample is small, both sample distributions free of strong skewness and outliers

t = [(x̄1 − x̄2) − Δ0] / √(s1²/n1 + s2²/n2); for H₀: μ1 − μ2 = 0, t = (x̄1 − x̄2) / √(s1²/n1 + s2²/n2)

Usually found with technology; df may be noninteger and falls between min(n1 - 1, n2 - 1) and n1 + n2 - 2. If using a t-table, use df = min(n1 - 1, n2 - 1) as a conservative choice