If one set of batteries lasts 2 weeks on average, the average number of sets in 40 weeks would be 20.
The Poisson distribution is exponential and the mean and variance have the same value=20.
For a normal approximation we want N(20,20), but we have to apply a continuity correction because Po(X>25) uses discreet X but a normal distribution (and its approximation) use a continuous random variable. The "bandwidth" for discreet random variables is 1 unit or X±0.5. Therefore we need to adjust for N: N(X>25.5). We also need to standardise for Z: (25.5-20)/√20=1.23 approx. (standard deviation is the square root of the variance). This means that the difference between X and the mean is 1.23 standard deviations away from the mean, and we can look that up in tables for normal distribution.
This value of Z corresponds to a probability of 0.8907; but this is the area to the left of Z (cumulative probability for less than 1.23), and we want the area to the right, so we need 1-0.8907=0.1093, which approximates to 11%. This means the probability of more than 25 sets of batteries is about 11%.