Calculating Type II error

A dean in the business school claims that GMAT scores of applicants to the school's MBA program have increased during the past 5 years. Five years ago, the mean and standard deviation of GMAT scores of MBA applicants were 580 and 50, respectively. 23 applications for this year's program were randomly selected and the GMAT scores recorded. If we assume that the distribution of GMAT scores of this year's applicants is the same as that of 5 years ago, find the probability of erroneously concluding that there is not enough evidence to supports the claim when, in fact, the true mean GMAT score is 610. Assume α is 0.02.

Solve P(Type II Error):

I'm not sure if I have the formula mistaken, but I can't seem to get this question right. I either end up with a probability that doesn't make sense (i.e # > 1) or probabilities very skewed like .0002 or .9998. Any help would be great here.

The answer is $1-p$ where $p$ is the probability of indeed rejecting the null hypothesis that there is no increase. For that, just use the fact that this year's (true) mean is 610, and the standard deviation is the same (50), and you have $23$ samples, you want to find the probability that this year's empirical mean appears higher than 580, to a p-value of $0.02$ or lower, which is equivalent to finding $P(X+(610-580))/(\sqrt{23} \cdot 50) \geq z)$, where $z$ is the scaled and translated critical value for which the standard normal distribution has probability $0.02$ of being greater than or equal to $z$, assuming mean $0$ and standard deviation $1$ for scaled and translated $X$, i.e. standard normal distribution.