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One of the exercises from my lecture gives us the number of the population (which are annual savings) and the population mean and standard deviation at time T. Then at time T + 1, a random sample is chosen from the population and the sample mean and sample standard deviation (which is the same as the population standard deviation at time A) are given. We are asked to test the substantial hypothesis that the annual savings have increased in time T+1, assuming a significance level of 1%.

The lecturer told us that we should use one-sample t-test for both cases (I currently cannot contact him). Why shouldn't we use t-test for two population means in this cases?

On the other hand, in another question in which every data in sample is listed out for us, we should use t-test for two population means despite the two sample standard deviations of the two samples from the two population in concern are not the same. The substantial hypothesis to be tested is: "Habitants in City A spends more money than in City B." In this question, a random sample of a fixed number of participants are selected in both cities and asked about the money they spent at where they live.

In another of the exercises, a random sample of people are selected to provide satisfaction index in time T and time T+1 respectively (i.e. the two sample (at the two different time point) are not independent). Every data in the sample is given. The hypothesis to be tested is: "The mean satisfaction of them has improved from T to T+1." One-sample t-test is used to test this hypothesis. Am I correct that t-test for two population means can not be used because the two samples are not independent of each other?

So how should we decide between using one-sample t-test and t-test for two population means?

This is our formula of t-statistic the t-test for two population means: $$t= \frac{\overline{x}-\overline{y}}{\sqrt{\frac{(n_1+n_2)[(n_1-1)s_1^{2}+(n_2-1)s_2^{2}]}{n_1\cdot n_2(n_1+n_2-2)}}}$$

$\overline{x}$ and $\overline{y}$: sample means

$n_1$ and $n_2$: sample size

$s_1$ and $s_2$: sample standard deviation

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In the first case, you have been given the population mean at time $T$, and, as a consequence, you only have one sample - the time $T+1$ sample - which you are comparing to a known parameter value. So you use the one-sample t-test. If, instead of being given the population mean and standard deviation at time $T$, you had been given a sample from that population, you would have two samples, one from time $T$ and one from time $T+1$, and you would use the two-sample t-test.

In the second case, you don't have a population mean to compare to, you only have two samples - one from City A and one from City B. Consequently, you use the two-sample t-test. If you had been given the actual population mean for, say, City A, then you would be comparing a sample from City B to a known value for City A and the one-sample test would be appropriate.

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  • $\begingroup$ One of the assumptions to use the t-test involving the formula I newly added to the question is that the variance of the value in the two population must be the same. How is this condition satisfied? (The two sample variance are not the same.) $\endgroup$
    – Aqqqq
    Commented Jan 22, 2018 at 7:05
  • $\begingroup$ In another of the exercises, a random sample of people are selected to provide satisfaction index in time T and time T+1 respectively (i.e. the two sample (at the two different time point) are not independent). Every data in the sample is given. The hypothesis to be tested is: "The mean satisfaction of them has improved from T to T+1." One-sample t-test is used to test this hypothesis. Am I correct that t-test for two population means can not be used because the two samples are not independent of each other? $\endgroup$
    – Aqqqq
    Commented Jan 22, 2018 at 7:11
  • $\begingroup$ Both your comments introduce a new question; please don't keep adding new questions in the comment thread, but rather post them as new questions. If you have questions that are about the answer, sure, but clearly these don't; the first relates to understanding a formula that was not in the original post, even implicitly, and the second relates to a new exercise that was not mentioned in the original post. $\endgroup$
    – jbowman
    Commented Jan 22, 2018 at 16:18
  • $\begingroup$ The formula is the supplemental information which is very relevant for the question. (I did not have time to add the formula and the question yesterday.) Note that my question is "So how should we decide between using one-sample t-test and t-test for two population means?", not "why should these two question be solved via one-sample t-test or t-test for two population means?" $\endgroup$
    – Aqqqq
    Commented Jan 22, 2018 at 16:25
  • $\begingroup$ You are now asking about the variance of the values of the two populations in relation to a specific formula, and you are doing so in a comment responding to the initial question, which was not about the variance of the values and did not contain the specific formula. That makes it a new question. Furthermore, I specifically tell you how you should decide. Read the first two sentences of each paragraph. They tell you specifically that if you have one sample that you are comparing to a known parameter value, you use the one-sample t-test, and... $\endgroup$
    – jbowman
    Commented Jan 22, 2018 at 16:31

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