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Consider a normally distributed random variable with an expected value of 20 and standard deviation of 4. If you determine the value of this variable 8 times, then what is the chance that the 8 sample values yield an improved standard deviation between 4.9 and 5.3?

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obs = 10000 # rows
tests = matrix(0, nrow = obs, ncol = 8) # 8 cols = 8 runs 

set.seed(1003)
# populate with random values, mean = 20, sd = 4
for (i in 1:obs) {
    for (j in 1:8) {
        tests[i,j] <- rnorm(1, mean = 20, sd = 4)
    }
}

#Find sd of each row
sds = apply(tests, 1, sd)

# Filled Density Plot
d <- density(sds)
plot(d, main="Kernel Density of Std. Dev.")
polygon(d, col="red")
abline(v = 4.9, col= "black")
abline(v = 5.3, col= "black")


# Counts for standard deviation between 4.9 and 5.3
counts = vector(mode =  "integer", length = obs)
for (k in 1:obs) {
    if ((sds[k] >= 4.9) & (sds[k] <= 5.3)) {
        counts[k] <- 1
    }
}
total = sum(counts)
print((sum(counts)/obs))

## [1] 0.0704

The percent probability of this event occurring is 7.04%.