It depends on your research question. Do you want to compare each sample mean to the "known" population mean of 60 (in which case the one-sample t test would be appropriate) or do you want to compare the two sample means against each other ( in which case you would use the independent samples t test)? It could very well be that each sample mean differs significantly from 60 but that the two sample means are not significantly different from one another. Hence the different results from the two procedures.

First thing: forget all decision on statistical methods based on sample size as the first criterion, it's typically 1) a very very old habit, dating from the before-computers era and 2) a very non-rigorous way to analyze data because it does not deal at all with the real important points: what is the question you wand to answer? How do you model your data? How do you use this model to answer this question? None of these points involves the sample size.

If you *know* (reasonnably) that your data are (close enough from a) normal (distribution), you do not have to bother with sample sizes, especially with such arbitrary rules like « more than 30 ».

Note that comparing t-tests on each sample individually (« one sample ») and a two-samples t-tests is totally meaningless, it shows that you simply did not really about « what is your question » because it answers two very, very different questions: the one-sample t-test answers to « is the theoretical mean from this sample different from 0 » (so it is not really surprising that the test is significant…) [by default; you can compare to another reference value, it does not change the logic], whereas the two-sample t-test answers to « do the two samples have the same theoretical mean » (which is probably your real question, and you can expect either a significant or a non-significant test; please keep in mind that « non-significant » does not mean « H0 is true/accepted », it only means « I cannot prove that H0 is false/reject H0 », which is completely different).

So I would really advise to take contact with a professional statistician locally to deal with your data. Or at least to follow a basic statistics course.

Hi, Christian Geiser I am comparing the means of each group (i.e. the one with 41 as the sample size and the other one one with 12) independently to the mean of 60. And then, I want to compare the two sample means to each other. This would require doing a one sample t test individually (i.e. the first group to the mean and then the second group to the mean) and then I would perform a two groups independent t test to compare them against each other. My last question is, would this be the correct procedure or would you recommend another procedure?

Where is this 60 value from? Does it have any error/interval associated with it?

Daniel Wright 60 is a score that I am comparing the sample means to

repeat: Where is this 60 value from? Does it have any error/interval associated with it?

Daniel Wright sorry for misinterpreting your question. It does not have any error or interval associate with it. Essentially I administered a survey with answers from 1-5 and was given to two groups (because there were two languages).

Ema Mejia: You asked: "Hi, Christian Geiser I am comparing the means of each group (i.e. the one with 41 as the sample size and the other one one with 12) independently to the mean of 60. And then, I want to compare the two sample means to each other. This would require doing a one sample t test individually (i.e. the first group to the mean and then the second group to the mean) and then I would perform a two groups independent t test to compare them against each other. My last question is, would this be the correct procedure or would you recommend another procedure?"

Yes, this sounds correct to me!

Hi dear Ema,

You need to define your goal.

1) Do you want to compare the average of each sample with the value of 60

or

2) Do you want to compare the averages of the two samples?

In the first case you should use a one-sample t-test and in the second case a t-test for independent samples. However, it depends on your research question.

Ema Mejia , maybe I am not making myself clear. Why have you chosen 60 to compare these means to? This may affect your choice for your analyses.

Also, you say the data are normally distributed. By that you presumably mean you are assuming they are drawn from a normal distribution. In general more information about what you did is necessary before at least I'd be comfortable recommending any particular analyses (e.g., 1) what were randomly sampling from ... 2) was this with a random number generator from a set list, 3) how were people allocated to the groups, etc.).

Daniel Wright I chose 60 because that was the value we were given to compare our sample means to. So I concluded with Christian's answer above that a one sample t test would be needed and comparing the two sample means to each other would require an independent samples t test. 1. randomly sampling from a variety of stakeholders (such as counselors, doctors, etc...) and 2) no, the aim was to get at least 30 samples for statistical significance when referring to a population, and 3)the allocation was basically, since our project being tested was available in two languages, those who knew one language would receive the survey in that language and those who knew the other language would receive the survey in the translated language.

If the means of a survey exceeded 60 then it meant that the product was satisfactory and easy to use.

What does "60 because that was the value we were given" mean? Is this a class assignment? I am going to unfollow this question. It may not be intentional but you are not answering any of my questions but just repeating the words.

Ema Mejia : if you really mean « exceed 60 », and not « is different from 60 », you should use unilateral tests. Did you do that? Beside, with your context, what was then the aim of comparing the two surveys?

Note that the answer to the second point suggets a very poor background to experimental design.

In general, you should really really take the time to discuss with a local, skilled statiscian and follow yourself a basic curse in statistics, not to learn a set of receipts and rules like « sample > 30 », but instead to understand the underlying logic of these tools, so that you can design meaningfull experiments and draw sounded conclusions... What you describe really looks like the use of "rules" without understanding the tools used, hence no serious conclusion can be drawn from them...

That's may sound a little rude answer, but for the long term, see it as a precious advice to make good science (whatever science it is).

The independent samples t-test is comparing the mean of one group to the other, that is comparing the mean of 80 to the mean of 88. It sounds like this is confused with testing against a mean of 60 in your second and last paragraphs.

To compare each group to a mean of 60, you use the one-sample t-test for each of the groups, as you mention in the beginning of your second paragraph.

One thing to consider: don't rely too much on p-values. It is helpful to also look at the size of the effect you are looking at. You could do this by plotting the two groups of data, perhaps as individual dots (https://r-graphics.org/R-Graphics-Cookbook-2e_files/figure-html/FIG-DISTRIBUTION-DOTPLOT-MULTI-1.png ) or as box plots ( https://www.dummies.com/wp-content/uploads/451447.image0.jpg ). Maybe add a line for the value of 60 you're comparing. These plots will probably tell the story.

EDIT:

I mocked up some data for this example, with one sample t-tests, two-sample t-tests, and plots. This can be run in R or at https://rdrr.io/snippets/ .

A = c(111.42, 116.38, 83.63, 69.86, 112.24, 15.28, 113.20, 76.95, 76.63, 51.23,

83.35, 160.69)

B = c(94.83, 100.81, 49.98, 112.68, 44.55, 66.74, 124.83, 91.95, 111.18, 138.23,

54.23, 17.87, 16.09, 152.20, 60.73, 104.82, 104.48, 73.51, 112.47, 167.87,

161.97, 145.30, 90.17, 99.65, 67.04, 13.52, 150.71, 81.03, 125.14, -15.21,

37.59, 117.49, 114.18, 138.43, 23.48, 102.70, 103.33, 27.73, 58.38,

157.91, 82.14)

t.test(A, B)

t.test(A, mu=60)

t.test(B, mu=60)

### Plot

Y = c(A, B)

Group = c(rep("A", length(A)), rep("B", length(B)))

Data = data.frame(Group, Y)

boxplot(Y ~ Group,

data=Data)

abline(h=60, col="blue", lwd=2)

library(ggplot2)

ggplot(Data, aes(x=Group, y=Y)) +

geom_dotplot(binaxis='y', stackdir='center') +

geom_hline(yintercept=60, color = "blue", size=1)

I think Daniel Wright nailed it: I think these questions concern a class assignment intended to help students understand the differences between one-sample and unpaired t-tests. If that is so, let me point out that all t-tests have a common format, as shown in the attached slides.

Re the unpaired (or independent samples) t-test, the most commonly used null hypothesis states that mu1-mu2 = 0. But it is possible to have a null hypothesis that specifies a non-zero difference between the two population means. Suppose, for example, that historical (near census) data tell us that the mean difference in height between men and women is 5". If we collected present day heights of men and women, we might wish to test a null hypothesis that states mu1-mu2 = 5, not 0. I.e., we might wish to test a null that states the male - female difference in height has not changed from the historically observed difference.

Admittedly, this is a bit of a contrived example. But I think it makes an important point that many/most users of t-tests don't know anything about. I also wish statistical software packages allowed users to specify non-zero differences between mu1 and mu2 when conducting an unpaired t-test. YMMV.

Bruce Weaver : R allows to specify any value under the NULL for both paired or unpaired T tests, in the t.test function, using the mu = ... parameter.

More usual examples of using T-tests with a non-0 H0 parameter include one of the usual procedure to perform equivalence tests (the one using two one-sided tests against the limits of the equivalence region), and the comparison of a slope to 1 in concordance studies.