Final answer:
The statement that estimators should have large variance is false; they should actually have low variance for more reliable estimates. Larger sample sizes improve the reliability and power of statistical tests, which is desirable.
Step-by-step explanation:
The statement that 'We want our estimators to have a large variance' is false. In statistical estimation, a desirable property of an estimator is that it has low variance, meaning that it produces values that are not too spread out from the mean and, therefore, more consistently close to the parameter it is estimating.
Smaller sample sizes lead to more variability in the estimation, and to capture the true population mean, we need to have a larger confidence interval. However, the goal is to have estimators with low variance because this reduces the overall uncertainty of an estimate.
When performing hypothesis testing, such as a test of two variances, one of the assumptions must be that the population distributions are normally distributed. If the null hypothesis is false, larger variance in the combined data indicates that there is a significant difference between the groups. In paired sample tests, two sample means from matched subjects are compared. Lastly, for testing hypotheses, a larger sample size generally provides a more reliable outcome by reducing errors and increasing the test power, which is desirable as it lowers the chances of not rejecting a false null hypothesis.