# Confidence Interval Information


## Confidence Intervals if $$\sigma$$ is Known

Point estimate $$\pm$$ EBM (Error bound for a population mean)

*EBM is also known as the "Margin of Error"

CL=Confidence Level

$$\alpha$$ = 1-CL

$$\frac{\alpha}{2} = \frac{1 - CL}{2}$$

We use the “Standard Normal Distribution” to calculate $$z_{\frac{\alpha}{2}}$$

To find $$z_{\frac{\alpha}{2}}$$ using Desmos:

inversecdf(normaldist(0,1), CL+ $$\frac{\alpha}{2}$$)

We are trying to capture the true population mean ($$\mu$$, this is a parameter) with this confidence interval!

## Confidence Intervals if $$\sigma$$ is Not Known

Use the “sample standard deviation” or $$s$$ instead.  Because of this, we have to use $$t$$ distributions.

$$\bar{x} \pm t_{\frac{\alpha}{2}}(\frac{s}{\sqrt{n}})$$

Point estimate $$\pm$$ EBM (Error bound for a population mean)

*EBM is also known as the “Margin of Error”

DF=Degrees of Freedom= $$n - 1$$

CL=Confidence Level

$$\alpha$$ = 1-CL

$$\frac{\alpha}{2} = \frac{1 - CL}{2}$$

To find $$t_{\frac{\alpha}{2}}$$ using Desmos:

inversecdf(tdist(Degrees of Freedom), CL+ $$\frac{\alpha}{2}$$ )

We are trying to capture the true population mean ($$\mu$$, this is a parameter) with this confidence interval!

## Confidence Intervals for Proportions

$$(\hat{p}(p$$ $$hat))$$ $$OR$$ $$p'(p$$ $$prime) =$$ sample proportion (think number of successes from Binomial Distributions)

If it’s wearing a “hat” it’s from a sample, not a population.  No “hat” then it’s a population parameter!

$$\hat{p} = \frac{x (number \; of \; successes)}{n (sample \; size)}$$

$$\hat{p} \pm z_{\frac{\alpha}{2}}\sqrt{\frac{\hat{p} \hat{q}}{n}}$$    or   $$\hat{p} \pm z_{\frac{\alpha}{2}}\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$$

Where   $$\hat{q} = 1 - \hat{p}$$

Point estimate $$\pm$$ EBP (Error bound for a population proportion)

*EBP is also known as the “Margin of Error”

CL=Confidence Level

$$\alpha$$ = 1-CL

$$\frac{\alpha}{2} = \frac{1 - CL}{2}$$

We use the “Standard Normal Distribution” to calculate $$z_{\frac{\alpha}{2}}$$

To find $$z_{\frac{\alpha}{2}}$$ using Desmos:

inversecdf(normaldist(0,1), CL+$$\frac{\alpha}{2}$$ )

We are trying to capture the true population proportion ($$p$$, this is a parameter) with this confidence interval!

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