07 Confidence Intervals Free Online FlashCards

What is a confidence interval?

A confidence interval is a range of plausible values for an unknown population parameter, combining a point estimate with sampling uncertainty.

What estimates the population mean?

The sample mean $\bar{x}$ is the point estimate for the population mean $\mu$.

What does a 95% confidence level mean?

A 95% confidence procedure produces intervals containing the true parameter in approximately 95% of repeated samples using the same method.

How does higher confidence affect interval width?

Increasing the confidence level increases the critical value and makes the interval wider when sample size and variability remain unchanged.

How is the margin of error calculated?

The margin of error is the critical value multiplied by the standard error: $\text{ME}=\text{critical value}\times\text{SE}$.

What is the mean CI formula when $\sigma$ is known?

When $\sigma$ is known, use $\bar{x}\pm z^*\frac{\sigma}{\sqrt{n}}$.

What is the mean CI formula when $\sigma$ is unknown?

When $\sigma$ is unknown, use $\bar{x}\pm t^*\frac{s}{\sqrt{n}}$ with $df=n-1$.

Why use the $t$-distribution when $\sigma$ is unknown?

The $t$-distribution has heavier tails than the standard normal because estimating $\sigma$ adds uncertainty.

How is a sample proportion calculated?

The sample proportion is $\hat{p}=x/n$, where $x$ is the number of successes and $n$ is the sample size.

What is the normal CI formula for a proportion?

The normal-approximation interval is $\hat{p}\pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

How is sample size planned for a mean?

Use $n=\left(\frac{z^*\sigma}{E}\right)^2$, then round up to the next whole observation.

What value of $p^*$ is conservative without prior information?

If no prior estimate is available, set $p^*=0.50$; it maximizes $p^*(1-p^*)$ and therefore gives the largest required sample size.