Distribution of $X_i$ | Sample size $n$ | Variance $sigma^2$ | Statistic | $small 1-alpha$ confidence interval |
$X_isimmathcal{N}(mu, sigma)$ | any | known | $displaystylefrac{overline{X}-mu}{frac{sigma}{sqrt{n}}}simmathcal{N}(0,1)$ | $left[overline{X}-z_{frac{alpha}{2}}frac{sigma}{sqrt{n}},overline{X}+z_{frac{alpha}{2}}frac{sigma}{sqrt{n}}right]$ |
$X_isim$ any distribution | large | known | $displaystylefrac{overline{X}-mu}{frac{sigma}{sqrt{n}}}simmathcal{N}(0,1)$ | $left[overline{X}-z_{frac{alpha}{2}}frac{sigma}{sqrt{n}},overline{X}+z_{frac{alpha}{2}}frac{sigma}{sqrt{n}}right]$ |
$X_isim$ any distribution | large | unknown | $displaystylefrac{overline{X}-mu}{frac{s}{sqrt{n}}}simmathcal{N}(0,1)$ | $left[overline{X}-z_{frac{alpha}{2}}frac{s}{sqrt{n}},overline{X}+z_{frac{alpha}{2}}frac{s}{sqrt{n}}right]$ |
$X_isimmathcal{N}(mu, sigma)$ | small | unknown | $displaystylefrac{overline{X}-mu}{frac{s}{sqrt{n}}}sim t_{n-1}$ | $left[overline{X}-t_{frac{alpha}{2}}frac{s}{sqrt{n}},overline{X}+t_{frac{alpha}{2}}frac{s}{sqrt{n}}right]$ |
$X_isim$ any distribution | small | known or unknown | Go home! | Go home! |
Distribution of $X_i$ | Sample size $n$ | Mean $mu$ | Statistic | $small 1-alpha$ confidence interval |
$X_isimmathcal{N}(mu,sigma)$ | any | known or unknown | $displaystylefrac{s^2(n-1)}{sigma^2}simchi_{n-1}^2$ | $left[frac{s^2(n-1)}{chi_2^2},frac{s^2(n-1)}{chi_1^2}right]$ |
Distribution of $X_i, Y_i$ | Sample size $n_X, n_Y$ | Variance $sigma_X^2, sigma_Y^2$ | Test statistic under $H_0$ |
Normal | any | known | $displaystylefrac{(overline{X}-overline{Y})-delta}{sqrt{frac{sigma_X^2}{n_X}+frac{sigma_Y^2}{n_Y}}}underset{H_0}{sim}mathcal{N}(0,1)$ |
Normal | large | unknown | $displaystylefrac{(overline{X}-overline{Y})-delta}{sqrt{frac{s_X^2}{n_X}+frac{s_Y^2}{n_Y}}}underset{H_0}{sim}mathcal{N}(0,1)$ |
Normal | small | unknown with $sigma_X=sigma_Y$ | $displaystylefrac{(overline{X}-overline{Y})-delta}{ssqrt{frac{1}{n_X}+frac{1}{n_Y}}}underset{H_0}{sim}t_{n_X+n_Y-2}$ |
Distribution of $X_i, Y_i$ | Sample size $n=n_X=n_Y$ | Variance $sigma_X^2, sigma_Y^2$ | Test statistic under $H_0$ |
Normal, paired | any | unknown | $displaystylefrac{overline{D}-delta}{frac{s_D}{sqrt{n}}}underset{H_0}{sim}t_{n-1}$ |
Coefficient | Estimate | $sigma$ | Statistic | $1-alpha$ confidence interval |
$alpha$ | $A$ | known | $frac{A-alpha}{sigmasqrt{frac{1}{n}+frac{overline{X}^2}{S_{XX}}}}simmathcal{N}(0,1)$ | $left[A-z_{frac{alpha}{2}}sigmasqrt{frac{1}{n}+frac{overline{X}^2}{S_{XX}}},A+z_{frac{alpha}{2}}sigmasqrt{frac{1}{n}+frac{overline{X}^2}{S_{XX}}}right]$ |
$beta$ | $B$ | known | $frac{B-beta}{frac{sigma}{sqrt{S_{XX}}}}simmathcal{N}(0,1)$ | $left[B-z_{frac{alpha}{2}}frac{sigma}{sqrt{S_{XX}}},B+z_{frac{alpha}{2}}frac{sigma}{sqrt{S_{XX}}}right]$ |
$alpha$ | $A$ | unknown | $frac{A-alpha}{ssqrt{frac{1}{n}+frac{overline{X}^2}{S_{XX}}}}sim t_{n-2}$ | $left[A-t_{frac{alpha}{2}}ssqrt{frac{1}{n}+frac{overline{X}^2}{S_{XX}}},A+t_{frac{alpha}{2}}ssqrt{frac{1}{n}+frac{overline{X}^2}{S_{XX}}}right]$ |
$beta$ | $B$ | unknown | $frac{B-beta}{frac{s}{sqrt{S_{XX}}}}sim t_{n-2}$ | $left[B-t_{frac{alpha}{2}}frac{s}{sqrt{S_{XX}}},B+t_{frac{alpha}{2}}frac{s}{sqrt{S_{XX}}}right]$ |
Sample size | Standardized statistic | $1-alpha$ confidence interval for $rho$ |
large | $displaystylefrac{V-frac{1}{2}lnleft(frac{1+rho}{1-rho}right)}{frac{1}{sqrt{n-3}}}underset{ngg1}{sim}mathcal{N}(0,1)$ | $displaystyleleft[frac{e^{2V_1}-1}{e^{2V_1}+1},frac{e^{2V_2}-1}{e^{2V_2}+1}right]$ |
# get means for variables in data frame mydata
# excluding missing values
sapply(mydata, mean, na.rm=TRUE)
# mean,median,25th and 75th quartiles,min,max
summary(mydata)
# Tukey min,lower-hinge, median,upper-hinge,max
fivenum(x)
library(Hmisc)
describe(mydata)
# n, nmiss, unique, mean, 5,10,25,50,75,90,95th percentiles
# 5 lowest and 5 highest scores
library(pastecs)
stat.desc(mydata)
# nbr.val, nbr.null, nbr.na, min max, range, sum,
# median, mean, SE.mean, CI.mean, var, std.dev, coef.var
library(psych)
describe(mydata)
# item name ,item number, nvalid, mean, sd,
# median, mad, min, max, skew, kurtosis, se
library(psych)
describe.by(mydata, group,...)
library(doBy)
summaryBy(mpg + wt ~ cyl + vs, data = mtcars,
FUN = function(x) { c(m = mean(x), s = sd(x)) } )
# produces mpg.m wt.m mpg.s wt.s for each
# combination of the levels of cyl and vs