Confidence Interval Information
- Page ID
- 1328
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Confidence Intervals if \( \sigma \) is KnownPoint estimate \( \pm \) EBM (Error bound for a population mean)
*EBM is also known as the "Margin of Error"
CL=Confidence Level
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!
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Confidence Intervals if \( \sigma \) is Not KnownUse 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
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!
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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
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! |
\(\alpha\) = 1-CL
\( \frac{\alpha}{2} = \frac{1 - CL}{2} \)by Katryn Weston