5 Estimation and Confidence Intervals Free Online FlashCards

What is a point estimate?

A point estimate is a single value calculated from sample data to estimate an unknown population quantity, such as a mean or proportion.

What is the general form of a confidence interval?

A confidence interval is the estimate plus or minus the margin of error: $\text{estimate} \pm (\text{critical value})(\text{standard error})$.

How do sample size and confidence level usually affect interval width?

Larger samples usually reduce standard error and narrow intervals; higher confidence levels require larger critical values and widen intervals.

What does 95% confidence mean?

A 95% confidence procedure captures the true parameter in about 95% of intervals over repeated samples, assuming the procedure’s conditions hold.

Which interval estimates a mean when population standard deviation is unknown?

With unknown population standard deviation, use $\bar{x} \pm t_{1-\alpha/2,\,n-1}\frac{s}{\sqrt{n}}$. The data should be reasonably well behaved, especially for small samples.

What replaces the t-based standard error when population standard deviation is known?

When the population standard deviation $\sigma$ is known, use the critical value $z_{1-\alpha/2}$ and standard error $\frac{\sigma}{\sqrt{n}}$.

How is a sample proportion calculated?

For $x$ successes in $n$ observations, the sample proportion is $\hat p=\frac{x}{n}$. A common large-sample interval uses standard error $\sqrt{\frac{\hat p(1-\hat p)}{n}}$.

When may a Wilson interval be preferable to the normal approximation?

The normal-approximation interval can perform poorly with few successes or failures, or with proportions near 0 or 1. A Wilson interval is often preferable in these cases.

How should paired data be handled when comparing means?

For paired data, analyze the within-pair differences. A confidence interval for a difference in means that contains zero does not prove the means are equivalent.

What should guide a confidence interval for a difference in proportions?

Estimate the difference $p_1-p_2$ with a method suited to the sample sizes and observed proportions; simple normal approximations may be inaccurate.

What does a regression slope confidence interval estimate?

A slope interval estimates the change in the mean outcome associated with a one-unit increase in a predictor, under the fitted model.

Which distribution is used for intervals for variance or standard deviation?

For normally distributed data, confidence intervals for variance or standard deviation use the chi-square distribution rather than the mean’s t-interval.