I have two samples, coming from different populations. One sample has 8,000 records, a mean of 5 and a sd of 0.5 The second has 1,500 records, a mean of 7 and a sd of 1.5 The distributions are close to normal.
This is coming from the behaviour of two kind of devices, and I want to understand if the output of one is of higher quality than the other.
Can I apply a $t$-test here? What cautions should I have or which corrections/alternative test do I have?
Assuming your samples are independent, then Welch's t-test does seem to be appropriate here, since it appears you have unequal variances (but you can formally test this too if you want through Levene's Test for Equality of Variances).
That being said, since you have quite large samples from both device 1 and device 2, then you can appeal to the central limit theorem and use:
\begin{eqnarray*} Z & = & \frac{\bar{X}-\bar{Y}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\sim N(0,1)\\ \end{eqnarray*}
under the null hypothesis of equal means. Here, $\bar{X}$ and $\bar{Y}$ and sample means from device 1 and device 2, respectively and $s_i^2$ and $n_i$ are the sample variance and sample sizes from the ith device $i=1,2$. Note that in large sample inference, you don't need to concern yourself with unequal variances.
Then a 95% confidence interval for your estimate would be given by:
\begin{eqnarray*} \bar{X}-\bar{Y} & \pm & Z_{\alpha/2}\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}} \end{eqnarray*}
where $Z_{\alpha/2}$ is the upper $\alpha/2$ point of the standard normal distribution.
All this being said, I wholeheartedly agree with the answer provided by Stefan. These sample sizes are really large and he's provided sound advice that you should follow. You should focus on what is an important practical difference. Is a 0.0001 mean difference between device 1 and device 2 important to you? Is it still important if device 1 costs three times as much as device 2?
With such a huge sample size almost any slight differences in those two means will be declared significant. Instead, I would try to visualize your samples in different ways to learn more about the shape of the data.
Also how is "higher quality" defined by you? Does it mean that the mean outcomes should be different? Or does it perhaps apply more to the variances between the samples, e.g. less variation more desirable?
Here are some ideas how to visualize the data using R:
require(ggplot2)
require(gridExtra)
d1 <- data.frame(Y = rnorm(8000, 5, 0.5), X = "A")
d2 <- data.frame(Y = rnorm(1500, 7, 1.5), X = "B")
d <- rbind(d1, d2)
p1 <- ggplot(d, aes(Y, group = X)) + geom_density() + ggtitle("Density plot")
p2 <- ggplot(d, aes(X, Y)) + geom_boxplot() + ggtitle("Boxplot")
p3 <- ggplot(d, aes(X, Y)) + geom_violin() + ggtitle("Violin plot")
grid.arrange(p1, p2, p3, ncol = 1)
