R - Normal distribution with top 10% larger than specific value

Question

I need to simulate a roughly normally distributed sample in R of size 500. 10%, i.e., 50 of the values should be larger than 50, the rest should be below 50 but still larger than 0.

I'm kind of stuck... any help is highly appreciated!

Answer 1

There is a way to do this. First of all, take 502 samples:

x <- rnorm(502)

Now normalize the output so that the minimum is 0 and maximum is 1:

x <- x - min(x)
x <- x / max(x)

Since the 0 and 1 are not random, we remove them from the sample:

x <- x[-c(which.min(x), which.max(x))]

Now we multiply the result by whichever number causes 10% of the sample to be greater than 50. We can use optimize for this:

f <- function(a) abs(sum((a * x) > 50)/length(x) - 0.1)
x <- optimize(f, c(0, 100))$minimum * x

This gives us what appears to be a normally distributed sample:

hist(x)

And exactly 10% of the samples are above 50:

length(x)
#> [1] 500

sum(x > 50)
#> [1] 50

None of the samples are less than 0:

min(x)
#> [1] 6.299734

And the result is not statistically different from a normal distribution:

shapiro.test(x)
#> 
#>  Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.99769, p-value = 0.7275

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Addendum

Incidentally, if you only need a single sample, a quick alternative is:

set.seed(4)
x <- rnorm(500, 35.3, 12)

Here, x is normally distributed with a minimum value of 1.22 and exactly 50 elements are over 50.