Confidence Interval Information

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Mar 18, 2021
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- Confidence Interval Information
- Videos For Elementary Statistics

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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!

by Katryn Weston

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Confidence Intervals if $\sigma$ is Known

Confidence Intervals if $\sigma$ is Not Known

Confidence Intervals for Proportions

Confidence Intervals if σ \sigma is Known

Confidence Intervals if σ \sigma is Not Known

Confidence Intervals for Proportions

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Confidence Intervals if $\sigma$ is Known

Confidence Intervals if $\sigma$ is Not Known