Diogo Bastos - 9 months ago 58

R Question

I have this sample:

`x=c(92L, 9L, 38L, 43L, 74L, 16L, 75L, 55L, 39L, 77L, 76L, 52L,`

100L, 85L, 62L, 60L, 49L, 28L, 6L, 27L, 63L, 22L, 23L, 99L, 61L,

25L, 19L, 48L, 91L, 57L, 97L, 84L, 31L, 87L, 1L, 21L, 30L, 41L,

13L, 72L, 68L, 95L, 47L, 11L, 24L, 58L, 18L, 67L, 33L, 8L, 50L,

4L, 40L, 12L, 73L, 78L, 86L, 69L, 44L, 83L, 94L, 65L, 37L, 70L,

54L, 46L, 15L, 53L, 89L, 98L, 90L, 3L, 14L, 17L, 42L, 45L, 79L,

20L, 32L, 34L, 64L, 88L, 81L, 96L, 59L, 71L, 56L, 26L, 51L, 29L,

80L, 7L, 36L, 93L, 82L, 35L, 5L, 2L, 10L, 66L)

I want to calculate this probability:

`P(x) > Mean(x) + 3`

So I do this:

`mean(x) = 50.5 ; sd(x)=29.01`

I generate the density distribution and calculate my probability, which now is:

`P(x) > 53.5`

pnorm(53.5, mean=mean(x), sd=sd(x), lower.tail=FALSE)

If I want calculating using Standard Distribution:

`P(x)>(53.5) = P(z=(x-mean(x)/sd(x))) > ((53.5 - 50.5)/29.01) = P(z)>(3/29.01)`

pnorm(3/29.01149, mean=0, sd=1, lower.tail=FALSE)

But when I want to use the T-Student Distribution, how can I proceed?

Answer Source

It is more legitimate to use `t`

distribution here, as standard error is estimated from data.

```
pt(3 / sd(x), df = length(x) - 1, lower.tail = FALSE)
# [1] 0.4589245
```

We have `length(x)`

number of data, but also estimate 1 parameter (standard error), so the degree of freedom for t-distribution is `length(x) - 1`

.

There is not much difference compared with using normal distribution, though, given that `length(x)`

is 100 (which is large enough):

```
pnorm(3 / sd(x), lower.tail = FALSE)
# [1] 0.4588199
```