5 Hypothesis Test

5.1 mean inference (z,t-test)

5.1.1 Z-test

Note that there’s no built-in Z-test in R. Use third party package BSDA (the code above written by Gemini)

# install.packages("BSDA")
library(BSDA)

Loading required package: lattice


Attaching package: 'BSDA'

The following object is masked from 'package:datasets':

    Orange

x <- c(1,1,1,1,1,4,3,2,1,1,1,1,1,11,1,1)
z.test(x, sigma.x = 0.5, conf.level = 0.95)


    One-sample z-Test

data:  x
z = 16, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 1.755005 2.244995
sample estimates:
mean of x 
        2

You can also make implement from the formular: reference this passage: https://cran.r-project.org/web/packages/distributions3/vignettes/one-sample-z-test.html

5.1.2 t-test

x <- c(1,1,1,1,1,4,3,2,1,1,1,1,1,11,1,1)
y <- c(1,1,1,1,1,4,3,2,1,1,1,1,1,11,1,1)
t.test(x)


    One Sample t-test

data:  x
t = 3.1298, df = 15, p-value = 0.006884
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 0.6379832 3.3620168
sample estimates:
mean of x 
        2

t.test(x,y)


    Welch Two Sample t-test

data:  x and y
t = 0, df = 30, p-value = 1
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -1.845594  1.845594
sample estimates:
mean of x mean of y 
        2         2

t.test(x,y, mu = 2, alter = "two.sided", var.equal = T)


    Two Sample t-test

data:  x and y
t = -2.2131, df = 30, p-value = 0.03464
alternative hypothesis: true difference in means is not equal to 2
95 percent confidence interval:
 -1.845594  1.845594
sample estimates:
mean of x mean of y 
        2         2

5.2 variance inference

Try to type ?var.test to get the help document.

var.test(x,y)


    F test to compare two variances

data:  x and y
F = 1, num df = 15, denom df = 15, p-value = 1
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
 0.3493947 2.8620925
sample estimates:
ratio of variances 
                 1

5.3 goodness of fitness test

Test whether a sample data fit a distribution or not.

5.3.1 Normal Test

5.4 correlation

x <- c(1,1,1,1,1,4,3,2,1,1,1,1,1,11,1,1)
y <- c(1,1,1,1,1,4,3,2,1,1,1,1,1,11,1,1)
cor.test(x, y)


    Pearson's product-moment correlation

data:  x and y
t = Inf, df = 14, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 1 1
sample estimates:
cor 
  1

The default method, Pearson linear correlation is widely used.

5.5 median inference (non-para test)

Although it’s non-parametric test based on median, the function also require to pass parameter as mu.

x <- c(1,1,1,1,1,4,3,2,1,1,1,1,1,11,1,1)
y <- c(1,1,1,1,1,4,3,2,1,1,1,1,1,11,1,1)
wilcox.test(x)

Warning in wilcox.test.default(x): cannot compute exact p-value with ties


    Wilcoxon signed rank test with continuity correction

data:  x
V = 136, p-value = 0.0002424
alternative hypothesis: true location is not equal to 0

wilcox.test(x,y)

Warning in wilcox.test.default(x, y): cannot compute exact p-value with ties


    Wilcoxon rank sum test with continuity correction

data:  x and y
W = 128, p-value = 1
alternative hypothesis: true location shift is not equal to 0

wilcox.test(x,y, mu = 2, alter = "two.sided", var.equal = T)

Warning in wilcox.test.default(x, y, mu = 2, alter = "two.sided", var.equal =
T): cannot compute exact p-value with ties


    Wilcoxon rank sum test with continuity correction

data:  x and y
W = 33.5, p-value = 0.0001599
alternative hypothesis: true location shift is not equal to 2