Just some clarification regarding the central limit theorem. I understand that the theorem says
"Given a distribution with mean $\mu$ and variance $\sigma^2$, the sampling distribution of the mean approaches a normal distribution with mean $\mu$ and variance $\sigma^2/n$, as the sample size increases"
I am unclear as to how this extends to some real data examples.
For instance, say we are trying to estimate the average monthly electricity use for a household in London.
- The population is therefore households in London.
- The sample would be a subset of London properties.
Since we wish to estimate a monthly average, the data collected from each household would need to be a monthly figure. Therefore, looking at the distribution of household means, does the CLT say that this would approximate a normal distribution?
I have a sample of this data, which contains around 2000 properties. However, due to the nature of the data (cannot have electricity usage <0, and no upper limit), the distribution is heavily skewed. Therefore, I cannot see how the CLT can be applied?
The problem, is that I wish to obtain a confidence interval around the mean, and compare this with different sample sizes. I was planning on using a simple t-test, however this assumes normality.
Have I got some of my assumptions/logic wrong?