  Let $X$ be a set of items of size $n$ that contains some defective items,
denoted by $I$, where $I \subseteq X$. In group testing, a {\it test} refers to
a subset of items $Q \subset X$. The outcome of a test is $1$ if $Q$ contains
at least one defective item, i.e., $Q\cap I \neq \emptyset$, and $0$ otherwise.
  We give a novel approach to obtaining lower bounds in non-adaptive randomized
group testing. The technique produced lower bounds that are within a factor of
$1/{\log\log\stackrel{k}{\cdots}\log n}$ of the existing upper bounds for any
constant~$k$. Employing this new method, we can prove the following result.
  For any fixed constants $k$, any non-adaptive randomized algorithm that, for
any set of defective items $I$, with probability at least $2/3$, returns an
estimate of the number of defective items $|I|$ to within a constant factor
requires at least $$\Omega\left(\frac{\log n}{\log\log\stackrel{k}{\cdots}\log
n}\right)$$ tests.
  Our result almost matches the upper bound of $O(\log n)$ and solves the open
problem posed by Damaschke and Sheikh Muhammad [COCOA 2010 and Discrete Math.,
Alg. and Appl., 2010]. Additionally, it improves upon the lower bound of
$\Omega(\log n/\log\log n)$ previously established by Bshouty [ISAAC 2019].
