cdf.StudentsT: Evaluate the cumulative distribution function of a StudentsT...

Description Usage Arguments Value See Also Examples

View source: R/StudentsT.R

Description

Evaluate the cumulative distribution function of a StudentsT distribution

Usage

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## S3 method for class 'StudentsT'
cdf(d, x, ...)

Arguments

d

A StudentsT object created by a call to StudentsT().

x

A vector of elements whose cumulative probabilities you would like to determine given the distribution d.

...

Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

A vector of probabilities, one for each element of x.

See Also

Other StudentsT distribution: pdf.StudentsT(), quantile.StudentsT(), random.StudentsT()

Examples

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set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

alexpghayes/distributions documentation built on April 8, 2021, 5:55 a.m.