The z-test tests the hypothesis H₀: μ = μ₀ against μ ≠ μ₀ when the standard deviation σ is known.

Formula for Z Score:

z = √n(x - x0)/σ

> x <- read.csv("tp.txt",header=F)
> x <- x[1:100,]
> z <- sqrt(100) * (mean(x) - 0)/sd(x)
> z

[1] -0.2334861

Calculate P value:

> p <- 2 * pnorm(-abs(z),0,1)
> p

[1] 0.8153839

Since p>0.05, we accept the hypothesis.


We then hypothesize water volume will not change under higher temperature at 80 degrees. So we checked 100 bottles of drinking water with 300 ml volume, and recorded the volume difference from 300 ml. We will test the Hypothesis H₀: μ = 0 against μ ≠ 0.

Data in "tp.txt":

> x <- read.csv("tp.txt",header=F)
> x <- x[1:100,]
> z <- sqrt(100) * (mean(x) - 0)/sd(x)
> z

[1] 18.28636

Calculate P value:

> p <- 2 * pnorm(-abs(z),0,1)
> p

[1] 1.06279e-74

Since p < 0.05, the hypothesis is rejected.