8.9 Confidence intervals: summary of formulas

$(\text{lower value},\text{upper value})=(\text{point estimate}-\text{margin of error},\text{point estimate}+\text{margin of error})$

$\text{margin of error}=\text{upper value}-\text{point estimate}$ OR $\text{margin of error}=\frac{\text{upper value}-\text{lower value}}{2}$

Use the Normal Distribution for Means $\text{ME}=z_{\frac{\alpha }{2}}\cdot \frac{\sigma }{\sqrt{n}}$

The confidence interval has the format $(\overline{x}-\text{ME},\overline{x}+\text{ME})$ .

Use the Student's-t Distribution with degrees of freedom $\text{df}=n-1$ . $\text{ME}=t_{\frac{\alpha }{2}}\cdot \frac{s}{\sqrt{n}}$

Use the Normal Distribution for a single population proportion $\hat{p}=\frac{x}{n}$

$\text{ME}=z_{\frac{\alpha }{2}}\cdot \sqrt{\frac{\hat{p}\cdot \hat{q}}{n}}\hat{p}+\hat{q}=1$

The confidence interval has the format $(\hat{p}-\text{ME},\hat{p}+\text{ME})$ .

$\overline{x}$ is a point estimate for $\mu$

$\hat{p}$ is a point estimate for $\rho$

$s$ is a point estimate for $\sigma$

$\mathrm{z*}$ is a critical value, based on the Normal Curve for an exact confidence interval such as for a 95% confidence interval. ${\mathrm{z*\; =\; 1.96}}^{}$

$\mathrm{t*}$ is a critical value, based on the Student's-t for an exact confidence interval.