# Simple Question regarding t-tests

I have the mean from a previous study (the Brief Symptom Inventory) that I would like to compare the mean found in my study. Both samples have a population over 100. I do not have the entire data set for the previous study. I typically use SPSS for my stats. I was hoping to use a paired sample t-test as I am comparing the mean scores of the same scale. However, I was told by the program this was not possible with the values I entered. Is there another way that I may do this? Basically, looking to compare the means to determine if there is a statistical difference without having the entire population.

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Thank you everyone for your help. This was a great start to send me in the right direction. –  user49003 Jun 25 at 23:21
Thank you again for all of your help. I should say that (this may not surprise you!) that I don't have a strong background in statistics. My background is in law and started my PhD and am learning stats "as I go". That said, I'm enjoying it. –  user49003 Jun 26 at 19:20
Here is a table of the sample: –  user49003 Jun 26 at 19:21
College male (N=143) 1.17 (.77) 1.02 (.79) .97 (.83) .47 (.49) .99 (.83) .89 (.69) .79 (.67) .29 (.27) I have done some google searches, using my text, other stats sources I have found useful but am still looking for the process - I would like to try to simulate the data if possible (spss, stata or even excel if this is possible). I appreciated the following post (posted in next comment) which is nearly exactly the same and was hoping that someone may be kind enough to post any helpful tips (or sites) that would guide me. –  user49003 Jun 26 at 19:26

You cannot do a paired sample t-test because you do not have paired samples. In fact, you only have one sample and one number.

What you can do is a one sample t-test, but, rather than compare to 0 (the typical default) you compare to the earlier mean (treating that as a fixed value). One way to do this would be to subtract the earlier mean from every score and then do a t-test vs. 0; but SPSS may make it easier (I am not an SPSS user).

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Thank you so much for your help. I shoud add, I use Stata as well, but am not as familiar with the program. –  user49003 Jun 25 at 23:17

Do you have the standard deviation and sample size from the previous study as well?

There are a few options (but as others have mentioned, you don't have paired data, so a paired t test is not one of them).

You could treat the previous study as the population/process and do a one-sample test on your data using the previous study mean as the null value. This ignores any uncertainty in the previous study and opens you up to criticism. If this is for your own personal interest and you will not be publishing or making decisions based on the outcome then this could be a reasonable approach.

You could use SPSS to calculate the mean and standard deviation of your data, then plug those numbers along with the mean, SD, and sample size from the prior study into the formula for the two-sample t-test and calculate it by hand.

You could simulate data from a normal distribution and make it have the same mean, SD, and sample size as the prior study, then put this data into SPSS as one sample, your data as the other sample, then have SPSS compute the 2 sample t-test for you from the data. I believe that SPSS has tools to simulate data (it has been a long time since I used SPSS), I don't know if it will let you specify the resulting mean and SD, if not you will need to use the transformation tools to get the mean and SD of the prior study.

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Thanks for your help! Yes, I do have the sample size and standard deviation. I would prefer not to select your first option as I would like to publish the data. –  user49003 Jun 25 at 23:19
I will do a search to inquire as to simulating data in SPSS. I do you Stata as well so can look at that program as well. –  user49003 Jun 25 at 23:21

You don't need the entire sample, but you do need the sample variance (and size).

Let's call $X$ the previous sample, $Y$ your sample. If you know sample mean $\bar X$, size $N_X$, and variance $S^2_X$ of the previous sample, and if you can assume equal variance, then you can calculate the $t$ statistic: $$\frac{\bar X-\bar Y}{S_p\sqrt{\frac{1}{N_X}+\frac{1}{N_Y}}}\sim t_{N_X+N_Y-2},\quad\quad S_p=\sqrt{\frac{(N_X-1)S^2_X+(N_Y-1)S^2_Y}{N_X+N_Y-2}}$$ For example, if you know that the previous BSI is $1.32\pm 0.72$, where $1.32$ is the sample mean and $0.72$ is the sample standard deviation, then $\bar X=1.32$ and $S^2_X=0.72^2$. If you know $N_X$ (you say that it is "over 100" - do you know the exact size?), a pocket calculator and a $t$ table can give you the answer.

Neither Ronald Fisher nor William Gosset were SPSS users ;-)

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Thank you! Yes, I have the exact sample size and standard deviation. –  user49003 Jun 25 at 23:17

Why not do an 2-sample independent $t$-test? Assign a value $\mu_{1}$ for the mean of your Brief System Inventory and another value $\mu_{2}$ for the mean your study. Or, alternatively, do a one-sample $t$ test on $\mu_{2} - \mu_{1}$ centered around 0.

You can't do a paired $t$-test for this main reason: there is not an explicit pairing between your data sets, which would imply that the sample sizes are equal (which clearly isn't the case).

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As I read it, he doesn't have the data for a 2 sample test. –  Peter Flom Jun 25 at 20:30