# T distribution - basic question I think

#### Bh78

##### New Member
Hello,

Is it possible to use a t test to test whether an individual observation is from the same population as a (small) sample?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Hello,

Is it possible to use a t test to test whether an individual observation is from the same population as a (small) sample?

I did not understand the last part of your question "... as a ..." It might help if you describe your question and its context. You would not have a mean with only one observation?

#### Bh78

##### New Member
I did not understand the last part of your question "... as a ..." It might help if you describe your question and its context. You would not have a mean with only one observation?
If it is possible to say that a single observation is unlikely to be from a population by considering the z score. I just wondered if it was possible to make a similar comment on a single observation when you only have a small sample?

#### Dason

You would not have a mean with only one observation?
Of course you would. It's pretty easy to take the mean of one observation.

If it is possible to say that a single observation is unlikely to be from a population by considering the z score. I just wondered if it was possible to make a similar comment on a single observation when you only have a small sample?
If you assume that the two groups have equal variances and are normally distributed then it would be feasible to do a t-test to test what you're interested in.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
I guess relooking at this Dason may be alluding to a one sample ttest

#### Dason

Not really. It would still be a two-sample t-test. You wouldn't want to treat this as a one-sample t-test where you test the sample with more than observation against a "mean" of whatever was in the sample with just one observation because you would be treating that second value as having no variation.

Code:
> # generate some fakedata
> # both sets come from the same distribution
> y.grp1 <- rnorm(20, 0, 1)
> y.grp2 <- rnorm(1, 0, 1)
> y <- c(y.grp1, y.grp2)
>
> y.grp1
[1]  0.649044532 -2.256491187  1.097277554 -1.254684115  0.156735591
[6]  0.272097697  1.142179114  0.095524117  1.612587487 -0.862135249
[11]  0.044291385  0.004164832 -0.569598917 -0.314329979 -1.853036794
[16] -0.517776731  0.331797174 -1.175629448  0.149509153  1.507455853
> y.grp2
[1] 0.6222212
>
> grp <- rep(c("group 1", "group 2"), c(20, 1))
>
> # Two sample t-test - this is what would be appropriate here.
> t.test(y ~ grp, var.equal = TRUE)

Two Sample t-test

data:  y by grp
t = -0.6614, df = 19, p-value = 0.5163
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-2.953803  1.535259
sample estimates:
mean in group group 1 mean in group group 2
-0.0870509             0.6222212

> # One sample t-test where you treat the value in the second group
> # as the mean you're testing against
> # This is what I'm saying you shouldn't do.
> t.test(y.grp1, mu = y.grp2)

One Sample t-test

data:  y.grp1
t = -3.0309, df = 19, p-value = 0.006874
alternative hypothesis: true mean is not equal to 0.6222212
95 percent confidence interval:
-0.5768476  0.4027459
sample estimates:
mean of x
-0.0870509

#### hlsmith

##### Less is more. Stay pure. Stay poor.
But the second observation doesn't have any variation. Unless you assume a distribution?

What does the "rep" do?

You would make the conclusion you failed to reject that it came from a different distribution because the 95% CI contains "0"?

Still feels a little wonky, since you don't know the distribution of group 2.