Statistical analysis on confidence intervals I have a data set where the data, when plotted, is not normal. Log-transforming the data makes it normal. Should confidence intervals for the population mean and hypotheses testing about the population mean, be conducted using the data on the original or on the log-transformed scale?
 A: Where there are data, there usually is a research question. Let's say you have data on healthcare costs in the past month for a random sample of 100 migraine patients. Your research question is: What is the true average healthcare cost for all migraine patients represented by the random sample of 100 patients?
When you plot the healthcare costs data for the 100 sample patients via a histogram, you notice they have a right-skewed distribution. However, when you plot the log-transformed healthcare costs, you notice they have an approximately normal distribution.  Thus, you decide to use the log-transformed data to estimate the true average healthcare cost.  Note that transforming/changing the data did NOT change your research question. The research question should NOT be altered after seeing the data. The reason we collected the data in the first place was to answer the original research question. 
As whuber pointed out, when you analyze the log-transformed data, you end up estimating the true average value of the log-transformed healthcare costs. How do you go from this quantity to the true average value of the (untransformed) healthcare costs? 
One way to deal with this is to proceed as follows:


*

*Draw a large number B of bootstrap samples (with replacement) of size n = 100 from the original sample; 

*For each bootstrap sample, fit an intercept-only simple linear regression model of the form:
log(healthcare cost) = beta0 + error 
and use the fitted model to obtain an estimate b0 of beta0 and an estimate s of the standard deviation sigma of the model errors.  

*For each bootstrap sample, compute the quantity exp(b0 + 0.5s^2), where s^2 represents the squared value of s. This quantity estimates the true average healthcare costs with the help of a so-called smearing factor. See, for instance, https://www.annualreviews.org/doi/pdf/10.1146/annurev.publhealth.28.082206.094100. 

*Use the appropriate quantiles of the bootstrap distribution of exp(b0 + 0.5s^2) to construct the desired 95% confidence interval for the true average healthcare cost. 
