How to calculate Weibull confidence interval using chi_square distribution/ I have a dataset, which I assume has a weibull distribution :
vec <- c(90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84,
         44, 59, 29, 118, 25, 156, 310, 76, 26, 44, 23, 62,
         130, 208, 70, 101, 208)

and want to know how to calculate the Confidence Interval for parameters scale and shape using chi_squared test.
I have calculated alpha(=scale). I also know how to just extract Confidence Interval from the packages such as fitdistrplus. I know the formula for Confidence Interval is
D(a) = 2[log L(a_hat) − log L(a)] ≤ χ2(α)

but I don't know how this formula works. Can somebody help?

All your entries in your dataset contribute to the likelihood function. If you maximize your likelihood function, then you will get the most likely values for your shape and scale parameters.
For that maximum value, the shape and scale can only have one specific value. Any other value will produce a lower value of the likelihood function.
Let's say you lower your likelihood function by a Chi-squared distribution, with 2 degrees of freedom, and a confidence interval of 90% (approx 1.35), then the shape and scale parameters have more "room" to take on a different value.
Basically, by lowering your likelihood function, you enable the shape and scale parameters to be other values, within a certain range.
See also:
https://reliawiki.org/index.php/Confidence_Bounds#Likelihood_Ratio_Confidence_Bounds
