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" )
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?