# Goodness of fit test for a normal distribuition

I have the following exercise that shows $$n=6$$ numbers:

$$1.40, 1.55, 1.35, 1.50, 1.29, 1.64$$

Is data normally distributed at the 5% significance level?

Surely $$\overline{x} = 1.455$$, $$s=0.1315674732$$ and $$n_1 = \cdots = n_6 = 1$$ but how can i continue?

It is difficult to judge normality with only $$n = 6$$ observations. Many small samples, regardless of origin, will "pass" popular normality tests unless there happens to be marked skewness or far outliers.

A Shapiro-Wilk test does not reject normality for your data. In R:

x = c(1.40,1.55,1.35,1.50,1.29,1.64)
shapiro.test(x)

Shapiro-Wilk normality test
data:  x
W = 0.97174, p-value = 0.9039

boxplot(x, horizontal=T, col="skyblue2")


This failure to reject normality is hardly a proof of normality. Below we see that samples of size six from uniform and exponential populations often "pass" tests of normality (with P-values above 5%).

set.seed(2019)
pv.u = replicate(10^5,shapiro.test(runif(6))$$p.value) mean(pv.u > .05) ## 0.93769 pv.e = replicate(10^5,shapiro.test(rexp(6))$$p.value)
mean(pv.e > .05)
[1] 0.78049