We consider the famous Rasch model, which is applied to psychometric surveys when n persons under test answer m questions. The score is given by a realization of a random binary (n,m)-matrix. Its (j,k)th component indicates whether or not the answer of the jth person to the kth question is correct. In the mixture Rasch model one assumes that the persons are chosen randomly from a population. We prove that the mixture Rasch model is asymptotically equivalent to a Gaussian observation scheme in Le Cam's sense as n tends to infinity and m is allowed to increase slowly in n. For that purpose we show a general result on strong Gaussian approximation of the sum of independent high-dimensional binary random vectors. As a first application we construct an asymptotic confidence region for the difficulty parameters of the questions.