Suppose I have a large population and I want to test if installing a new special light bulb can reduce energy consumption. Since I have a large population, I'll have people that usually consume high and some that consume less energy on a daily basis; therefore I would like to know the impact of the new special light bulb depending on their previous behavior. So I want to cluster my population into groups based on their energy consumption. Suppose that I get K groups after clustering. How can I determine the minimum sample size for each of my K+1 groups (+1 because I'm counting with the group that doesn't have the new light bulb, i.e, control group) to ensure some statistical rigor (for example some confidence level, etc..). I've only encountered literature about stratified sampling and cluster sampling, but none of those are what I want to do. In stratified sampling, they divide the population into K groups and then take elements from all of the groups to make sure every group will have elements in the (only) sample. In clustering sampling, they divide the population into K groups and only consider some of those groups for the sample.
EDIT 1: I found about one-way ANOVA: it can determine sample sizes for multiple groups but assumes as a null hypothesis that all of the groups have equal means (which is not the case here). In this case, the null hypothesis would have to be something like "there's no changing on the mean (energy consumption) on any of the groups with the new special light bulb". Even if one-way ANOVA could be used, I would have to give as an input a guess/estimation of the means for each group after getting the new special light bulb?
EDIT 2: I'd like to thank StatsStudent for his answer, which is very detailed and informative. However, his answer is not what I'm actually looking for. To be more detailed, I have the data on the energy consumption of the whole population (without the new special light bulb). And after clustering, I know there are K different types of behavior (for example, K=3: low, medium, and high energy consumption behaviors among the population). I want to know if installing the light bulb will affect the energy consumption of any of the K groups. Also, one aspect that would also like to include is to have more than one special light bulb, for example, three different light bulbs: A, B, and C. Can I use Multi-Way ANOVA to determine the sample size needed? I've read that this test is an extension of ANOVA, where there is more than one category, but only one category is of interest. The other category/categories are things that need to be controlled for (blocking/nesting/random effects/etc.). If I'm interested in determining if there is a difference in energy consumption due to light bulbs (A, B, and C) while controlling for energy behavior (low, medium, high) (K=3:groups from clustering) can and should I use two-way ANOVA? This will result in $3 \times 3 = 9$ groups.