Hypothesis testing is a method for using sample data to judge a claim about a population. You state a null hypothesis H₀ (no effect, no change) and an alternative hypothesis Hₐ (what you suspect). Then you ask how surprising the data would be if H₀ were true. Very surprising data count as evidence against H₀.
A confidence interval for a difference in means estimates μ₁ − μ₂, the gap between two population means. You build a two-sample t-interval: (x̄₁ − x̄₂) ± t* times the standard error √(s₁²/n₁ + s₂²/n₂). If the interval does not contain 0, the data show that the two means really differ.
A confidence interval for a difference in proportions estimates p₁ − p₂, the gap between two population proportions. You build a two-proportion z-interval: (p̂₁ − p̂₂) ± z* times the standard error. If the whole interval is above or below 0, the data show a real difference between the two groups.
A confidence interval for a mean gives a range of plausible values for a population mean μ. When σ is unknown, you build a one-sample t-interval: x̄ ± t* · s/√n, with df = n − 1. The sample mean x̄ is the center, and the margin of error t* · s/√n sets the width.
The t-distribution is a bell-shaped curve, centered at 0, with heavier tails than the standard normal curve. You use it instead of z when the population standard deviation σ is unknown and you estimate it with the sample standard deviation s. Its shape depends on the degrees of freedom, df = n − 1.
The margin of error is how far a sample estimate is likely to be from the true population value. It equals the critical value times the standard error: ME = z* · SE. For a proportion, that is z*√(p̂(1 − p̂)/n). Solving this formula for n tells you how large a sample you need.
A confidence interval for a proportion estimates a population proportion p from a random sample. The formula is p̂ ± z*√(p̂(1 − p̂)/n), where p̂ is the sample proportion and z* sets the confidence level. It is also called a one-proportion z-interval.
A confidence interval is a range of plausible values for a population parameter, built from sample data. It has the form point estimate ± margin of error. The confidence level, such as 95%, is how often the method captures the true value over many random samples.
A sampling distribution for a difference describes how p̂₁ − p̂₂ or x̄₁ − x̄₂ varies when you take two independent random samples. Its mean is the difference in the population values. Its standard deviation comes from adding the two variances, then taking the square root. With large enough samples, its shape is approximately normal.
The sampling distribution of a sample mean describes how the sample mean x̄ varies from one random sample to the next. Its mean is the population mean μ, and its standard deviation is σ/√n. It is normal when the population is normal, and approximately normal when n is at least 30.