Confidence Intervals in Statistics

Drawing conclusions about a population based on sample data.

An interval of values likely to contain a population parameter.

Probability a confidence interval contains the population parameter.

Sample proportion (𝑝̂) used to estimate population proportion (p).

Sample mean (x̄) used to estimate population mean (µ).

Probability a confidence interval does not contain the population parameter.

Using a point estimate and margin of error to create a range (LB, UB).

Selecting many samples to demonstrate how confidence intervals distribute around a parameter.

A range of values likely to contain the true population proportion, found by adding and subtracting the margin of error from the point estimate.

The maximum likely difference between the sample proportion (point estimate) and the true population proportion.

The sample proportion, used to estimate the population proportion (p).

We are 95% confident that the true population proportion lies within the calculated interval.

If a hypothesized value falls outside the confidence interval, we can reject the claim.

The sample must be random, and the sample size should be large enough to satisfy the conditions for normality (np̂ ≥ 5 and n𝑞̂ ≥ 5).

Point Estimate ± Z * √(𝑝̂(1-𝑝̂)/n)

A value from a standard distribution (Z or t) corresponding to a specific confidence level. It determines the margin of error.

Higher confidence level means a wider confidence interval and larger margin of error. We are more certain, but less precise.

Larger sample sizes lead to narrower confidence intervals and smaller margins of error, providing more precise estimates.

A single value used to estimate the unknown population parameter (p or µ) based on sample data.

Confidence intervals are expressed in two ways: a) (Lower Bound, Upper Bound), and b) Lower Bound < parameter < Upper Bound.

The formula for calculating the standard error of a sample proportion, used in confidence interval calculations for proportions.

A range of values likely to contain the true population proportion (p), based on a sample proportion (𝑝̂).

The distance between the point estimate (𝑝̂) and the bounds of the confidence interval (LB, UB), representing the uncertainty in the estimate.

The critical value from the standard normal distribution used to determine the margin of error for a specific confidence level.

Conditions for normality that must be met to use the confidence interval formula for proportions. At least 5 successes (𝑛𝑝̂) and 5 failures (𝑛𝑞̂) are needed in the sample.

A confidence interval for the population proportion means that we are [confidence level] confident that the true population proportion falls between the lower bound (LB) and the upper bound (UB) of the interval.

A sample that accurately reflects the characteristics of the population you want to study, ensuring the results can be generalized.

A probability distribution used to estimate population means when the population standard deviation is unknown. It has a bell-shaped curve like the z-distribution but with adjustments for smaller sample sizes.

The number of values in a sample that are free to vary. Calculated as n - 1, where n is the sample size.

The standard deviation of the t-distribution varies depending on the sample size (n), but it's always greater than 1. As n gets larger, the t-distribution approaches the z-distribution.

When the population standard deviation (σ) is unknown, the t-distribution is used to estimate the population mean (μ) because it accounts for the additional uncertainty introduced by estimating σ using the sample standard deviation (s).

A range of values likely to contain the true population mean (μ). It is calculated as the point estimate (sample mean x̄) plus or minus the margin of error.

The sample mean (x̄) is used as the best estimate of the population mean (μ).

A value from the t-distribution corresponding to a given confidence level and degrees of freedom (df). It determines the width of the confidence interval.

Larger sample sizes lead to narrower confidence intervals, meaning we're more certain about the population parameter.

Higher confidence level means a wider interval, increasing our guarantee of capturing the population parameter. But, it also reduces precision.

A point estimate is a single guess for the parameter based on the sample. A confidence interval uses the estimate and margin of error to provide a range of plausible values.

The margin of error represents the +/- range around the point estimate. It quantifies how uncertain we are about the true population parameter.

A confidence interval is a range of plausible values for the population parameter, with a stated confidence level. We are confident that interval includes the true value most of the time.

The Central Limit Theorem states that for large samples, the distribution of sample means approaches a normal distribution, even if the original data is non-normal.

Confidence intervals are used to estimate population parameters from sample data. They help us understand how much uncertainty surrounds our estimates.