Why does my tstatistic have a negative sign?
The t statistic may be positive or negative depending on the direction of the difference between the means of two independent groups (unpaired t), the difference in pairs of means (paired t) or if a value is negative or positive (one sample ttest assuming a single variable has a zero mean).
The sign of the tstatistic not relevant if we are interested in twotailed tests where the direction of difference is not important but if the direction is specified apriori then we need to remember the outputted pvalue is the probability of observing a difference at least as extreme as that observed in the sample.
Onetailed tests may be of two forms both with the same null hypothesis (denoted by H0):
H0: mean 1 = mean 2 vs either HA: mean 1 < mean 2 or HA: mean 1 > mean 2
For example, in an unpaired ttest, if the second mean is found to be higher than the first mean, and this is expected apriori, the onetailed pvalue is half the outputted twotailed pvalue in SPSS. If, however, the direction of difference between the means is the opposite to that expected apriori then the onetailed pvalue equals 10.5*(twotailed pvalue).
The default among statistical packages performing tests is to report twotailed pvalues. Because the most commonly used test statistic distributions (standard normal, Student's t) are symmetric about zero, most onetailed pvalues can be derived from the twotailed pvalues. The example below is taken from this UCLA article.
Below, we have the output from a twosample ttest in [the statistical package] Stata. The test is comparing the mean male score to the mean female score. The null hypothesis is that the difference in means is zero. The twosided alternative is that the difference in means is not zero. There are two onesided alternatives that one could opt to test instead: that the male score is higher than the female score (diff > 0) or that the female score is higher than the male score (diff < 0). In this instance, Stata presents results for all three alternatives. Under the headings Ha: diff < 0 and Ha: diff > 0 are the results for the onetailed tests. In the middle, under the heading Ha: diff $$\ne$$ 0 (which means that the difference is not equal to 0), are the results for the twotailed test.
Twosample t test with equal variances
Group 
Obs 
Mean 
Std. Error 
Std. Dev. 

Male 
91 
50.121 
1.080 
10.305 

Female 
109 
54.991 
0.779 
8.134 

Diff 

4.870 
1.304 
8.134 
Degrees of freedom: 198
H0: mean(male)  mean(female) = diff = 0
Ha: diff < 0 
diff $$\ne$$ 0 
diff > 0 
t = 3.7341 
t = 3.7341 
t = 3.7341 
P < t = 0.0001 
P > t = 0.0002 
P > t = 0.9999 
Note that the test statistic, 3.7341, is the same for all of these tests. The twotailed pvalue is P > t. This can be rewritten as P(>3.7341) + P(< 3.7341). Because the tdistribution is symmetric about zero, these two probabilities are equal: P > t = 2 * P(< 3.7341). Thus, we can see that the twotailed pvalue is twice the onetailed pvalue for the alternative hypothesis that (diff < 0). The other onetailed alternative hypothesis has a pvalue of P(>3.7341) = 1(P<3.7341) = 10.0001 = 0.9999. So, depending on the direction of the onetailed hypothesis, its pvalue is either 0.5*(twotailed pvalue) or 10.5*(twotailed pvalue) if the test statistic symmetrically distributed about zero.
In this example, the twotailed pvalue suggests rejecting the null hypothesis of no difference. Had we opted for the onetailed test of (diff > 0), we would fail to reject the null because of our choice of tails.