Validity of Hypothesis Testing for Non-Normal Data

Question

I have never been able to clearly understand the relationship between Hypothesis Testing, Central Limit Theorem and the Normal Distribution.

As an example, suppose we have the salaries of randomly sampled people in some town (I made sure that the data does not look normally distributed). Using the R programming language, I simulated some data and plotted the results:

set.seed(123)
a = rnorm(100000,20000,1000)

a2 = rnorm(100000,40000,10000)

a1 = rnorm(100000,100000,10000)



salary = c(a,a1, a2)



###plot 


plot_1 =  hist(salary,  100000, ylab = "Number of People" , xlab = " Salary ",   main="Salary of Randomly Selected People in Some Town" )

My Question:

• A researcher believes that 95% of the town earns less than $105,000.00.

• However, The 90th percentile of this sampled data is approximately: $105,000.00

   quantile(salary, probs = 0.9)
  
     90% 
        105281.6 
  

Can we use hypothesis tests to estimate whether 90% of the population in this town earn less $105,281.00?

Null Hypothesis: Ho = 0.95

Alternate Hypothesis: Ha < 0.95

Test Statistics : Z = (0.95 - 0.90) / sqrt(0.9*(1 - 0.9) / N)) = 0.05 / sqrt(0.09/300000) = 91.28

Based on this extremely large value of the the Test Statistic, I don't think I am doing something right? This large value of the Test Statistic would suggest you accept the Null Hypothesis and conclude with 95% statistical significance that 95% of the people in the town (i.e. true population) earn less than $105,000.00?

I have a feeling the test statistic is so large because I have a really large sample size? Does the non-normal shape of my distribution affect the hypothesis test? Should I be using some other type of hypothesis test (e.g. non-parametric) for this problem?

Thanks!

Reference:

1. https://stats.libretexts.org/Courses/Las_Positas_College/Math_40%3A_Statistics_and_Probability/08%3A_Hypothesis_Testing_with_One_Sample/8.04%3A_Hypothesis_Test_Examples_for_Proportions

2. https://online.stat.psu.edu/statprogram/reviews/statistical-concepts/proportions

Answer 1

Generally speaking, the answer is yes. The methodology of hypothesis testing isn't bound only for normal data, but can be used for other types as well (see an example here). However, You might want to formulate your hypotheses differently, e.g.

In my opinion, there are three ways in which you can approach this problem:

1. Non-parametric, sampling-based methods: Bootstraping percentile confidence intervals (chapter 13 of Efron & Tibshirani) or some kind of permutation testing (intro here).
2. Non-parametric, rank-based methods: Wilcoxon, Mann-Whitney, there are a few possibilities.
3. Going with the usual hypothesis testing methodology. Given a series $\hat{\theta}_n$ of consistent and asymptotically-normal estimators (that is, $\forall\theta\in\Theta:\sqrt{n}(\hat{\theta}_n-\theta)\overset{d}{\rightarrow}N\left(0,\frac{1}{I(\theta)}\right)$), according to Wilks' theorem the generalized LRT statistic $\lambda$ for the problem $H_0:\theta=\theta_0,H_a:\theta\neq\theta_0$ follows $2\log(\lambda)\overset{d}{\rightarrow}\chi^2$ under $H_0$. You can play with this idea in order to construct a test for the 95^th percentile of your data. You should, however, use this with extreme caution as the assumptions on $\hat{\theta}_n$ are not trivial.