### R Z-test

Formula for Z Score:

z = √n(x - x_{0})/σ

Where:

n: Sample number

x: Population mean

x_{0}: Hyposized population mean

σ: Standard Deviation

We hypothesize water volume will not change under X rays. 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 H_{0}: σ = 0 against σ ≠ 0.

Calculate the Z Score:

> 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 H

_{0}: σ = 0 against σ ≠ 0.

Data in "tp.txt":

Calculate the Z Score:

> 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.