Summary of Formulae for Confidence Intervals

By TARMath Published November 12, 2020 Mathematics, Statistics

Use the tool that you are most comfortable using. Formulae below use the following symbols: α = alpha = 1 – Confidence Level, σ = population standard deviation, s = sample standard deviation, μ = population mean, ${\bar {x}}$ = sample mean, n = sample size, x = a data value, p is the percent below, df = degrees of freedom. Note z_α/2 is the z-score for percent α/2, normally converted to positive, similarly with t_α/2.

Normal Distributions (Known Population Standard Deviation)
To find z-score given data values: 1. $LaTeX: z-score\;=\;\frac{x-\mu}\sigma$; 2. Desmos: https://www.desmos.com/calculator/szi3lxfwix use the Finding Z-score from Data portion; 3. The “84 Calculators”: type in formula above (don’t forget parentheses!); 4. Excel: =STANDARDIZE(x,μ,σ)

To find z-score given a percent/probability:: 1. Normal/t-distribution Probabilities; 2. Desmos: https://www.desmos.com/calculator/szi3lxfwix use the Finding Z-score from Percent/Probability portion; Or use Formula > Dist > normaldist(m,s).inversecdf(p) , if you want percent above, use 1 – p.; 3. The “84 Calculators”: 2nd > DIST > TI Distr > line 3 InvNorm(area[μ, σ]), where area is the percent/probability. The μ & σ are optional, without you have your z, with them you have your x.; 4. Excel for standardized z: =NORM.S.INV(p); 5. Excel for x: =NORM.INV(p,μ,σ)

Margin of Error For Mean with Known Population Standard Deviation, use the Normal Distribution: 1. $LaTeX: E=z_{\alpha/2}\ast\frac{\sigma}{\sqrt{n}}$; 2. Desmos: https://www.desmos.com/calculator/n2ne45e1t2 use the Margin of Error and Confidence Intervals folder.; 3. The “84 Calculators”: find the Confidence Interval below: then $LaTeX: \frac{high\:value-low\:value}{2}$; 4. Excel: =CONFIDENCE.NORM(α,σ,n), don’t be fooled, this is the ERROR not the Interval!
: Confidence Interval of mean, µ; 1. $LaTeX: \overline x\pm E,\;or\;(\overline x-E,\overline x\;+E)$; 2. Desmos: https://www.desmos.com/calculator/n2ne45e1t2 use the Margin of Error and Confidence Intervals folder.; 3. The “84 Calculators”: If you have the list, use STAT > TEST > line 7 ZInterval, then enter your σ, your list of numbers (tedious to do if they aren’t already in there), Freq (usually 1), & Confidence Level.; If you have the statistics , use STAT > TEST > line 7 ZInterval > click the Stats at the top, then enter your σ, ${\bar {x}}$ , n, & Confidence Level; 4. Excel: find the Margin of Error above, then add and subtract it from the sample mean.

Sample Size with Known Population Standard Deviation: 1. $LaTeX: n=\frac{\left(z_{\alpha/2}\right)^2\ast\sigma^2}{E^2}$; 2. Desmos: https://www.desmos.com/calculator/n2ne45e1t2 use the Sample Size Calculator for Z-scores folder.; 3. The “84 Calculators”: type in formula above (don’t forget parentheses!); 4. Excel: type in formula above (don’t forget parentheses!)

Student t-Distributions (Unknown Population Standard Deviation)
Degrees of Freedom, df = n – 1, needed for all t calculations
To find t-score given data values: 1. $LaTeX: t-score= \frac{(\overline x-\mu)}{ s/{\sqrt n}}$; 2. Desmos: https://www.desmos.com/calculator/boxffyc34q use the Finding t-score from Data portion; Or use Formula > Dist > tscore(list), where list is you list of data, either individually or as a set.; 3. The “84 Calculators”: type in formula above (don’t forget parentheses!); 4. Excel: type in formula above (don’t forget parentheses!)

To find t-score given a percent/probability:: 1. Normal/t-distribution Probabilities; 2. Desmos: https://www.desmos.com/calculator/boxffyc34q use the Finding t-score from Percent/Probability portion; Or use Formula > Dist > tdist(df).inversecdf(p), if you want percent above, use 1 – p.; 3. The “84 Calculators”: 2nd > DIST > TI Distr > line 4 InvT(area,df ), where area is the percent/probability .; 4. Excel for t: =T.INV(p,df)

Margin of Error For Mean with Unknown Population Standard Deviation, use the Student’s t-Distribution: 1. $LaTeX: E=t_{\alpha/2}\ast\frac{s}{\sqrt{n}}$; 2. Desmos: https://www.desmos.com/calculator/5dd4bi2rhs use the Margin of Error and Confidence Intervals folder.; 3. The “84 Calculators”: find the Confidence Interval below: then $LaTeX: \frac{high\:value-low\:value}{2}$; 4. Excel: =CONFIDENCE.T(α,s,n), don’t be fooled, this is the ERROR not the Interval!
: Confidence Interval of mean, µ; 1. $LaTeX: \overline x\pm E,\;or\;(\overline x-E,\overline x\;+E)$; 2. Desmos: https://www.desmos.com/calculator/5dd4bi2rhs use the Margin of Error and Confidence Intervals folder.; 3. The “84 Calculators”: If you have the list , use STAT > TEST > line 8 TInterval , then enter your list of numbers (tedious to do if they aren’t already in there), Freq (usually 1), & Confidence Level.; If you have the statistics , use STAT > TEST > line 8 TInterval > click the Stats at the top, then enter your ${\bar {x}}$ , s, n, & Confidence Level; 4. Excel: find the Margin of Error above, then add and subtract it from the sample mean.

For Proportions with np’>5 & nq’>5
If a population proportion or estimated proportion is not given, use proportion = .5: For this section, p’ is for proportion and p for percent/probability.; q’=1 – p’; The appropriate mean to use in proportion formulae is np, this is the mean from the Binomial Probability!; The appropriate standard deviation to use in proportion formulae is: $LaTeX: \sigma=\sqrt{np'q'}$

Margin of Error: 1. $LaTeX: E=z_{\alpha/2}\sqrt{\frac{p'q'}{n}}$; 2. Desmos: https://www.desmos.com/calculator/zfdl21jyma use the Margin of Error and Confidence Intervals folder.; 3. The “84 Calculators”: find the Confidence Interval below: then $LaTeX: \frac{high\:value-low\:value}{2}$; 4. Excel: use z-score from normal section and type in formula above (don’t forget parentheses!)
: Confidence Interval of mean, µ; 1. $LaTeX: p'\pm E,\;or\;(p'-E,p'+E)$; 2. Desmos: https://www.desmos.com/calculator/zfdl21jyma use the Margin of Error and Confidence Intervals folder.; 3. The “84 Calculators”: If you have the list, use STAT > TEST > line 11 1-PropZInt(x,n,C-Level), where x is the number of successes, n it the total number, and C-Level is the confidence Level; 4. Excel: find the Margin of Error above, then add and subtract it from the sample proportion.

Sample Size: 1. $LaTeX: n=\frac{\left(z_{\alpha/2}\right)^2\ast\left(p'q'\right)^2}{E^2}$; 2. Desmos: https://www.desmos.com/calculator/zfdl21jyma use the Sample Size Calculator for Proportions folder.; 3. The “84 Calculators”: type in formula above (don’t forget parentheses!); 4. Excel: type in formula above (don’t forget parentheses!)

Just a reminder The “84 Calculators” are either the Texas Instruments TI-84, or the apps based on it: Calc84 (Graphing Calculator plus 84 graph emulator free 83) by lethinhien for Android and Graphing Calculator Plus by Incpt.Mobis for Apple.

There are other tools out there, some are not always correct.

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