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Confidence Interval Information

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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

    Confidence Interval Information is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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