Evaluate Accuracy of estimated continuous random variable

[110699]

The question presents us with a pdf for the continuous random variable X, which represents the height of students (18 years old). "It is estimated that X possesses the probability density function of:

f(x) = 348x-x^2-29700/18432 for 150<x<198"

I have confirmed that the estimated pdf is, in fact, a pdf as the integral between 150 and 198 is equal to 1. I also found the estimated value (=174) and the variance (=115.2) by using their respective formulae.

The next part of the question is to "evaluate the accuracy of this estimated model by using a suitable emprirical test to evaluate the model." I have collected the following data:

186	178	198	172	179	181
188	196	181	189	183	191
183	201	170	187	187	175
186	190	180	189	186	175
175	187

From the data it is already evident that the heights recorded fall outside of the given domain of (150<x<198) from the estimated model.Would the next step be to compare the average height from the data to the expected value given by the estimated pdf? (pdf gives 174 vs 184.3 from the data set). I am a bit stuck at the moment and any guidance would be great!

The next part of the question is to use the data collected to construct a pdf of my own to represent X. I haven't approached this problem either so if someone could point me in the right direction it would be greatly appreciated.

 

[Ishuda]

...

The next part of the question is to "evaluate the accuracy of this estimated model by using a suitable emprirical test to evaluate the model." I have collected the following data:

186	178	198	172	179	181
188	196	181	189	183	191
183	201	170	187	187	175
186	190	180	189	186	175
175	187

From the data it is already evident that the heights recorded fall outside of the given domain of (150<x<198) from the estimated model.Would the next step be to compare the average height from the data to the expected value given by the estimated pdf? (pdf gives 174 vs 184.3 from the data set). I am a bit stuck at the moment and any guidance would be great!

The next part of the question is to use the data collected to construct a pdf of my own to represent X. I haven't approached this problem either so if someone could point me in the right direction it would be greatly appreciated. :grin::grin:

First: "evaluate the accuracy of this estimated model...". What have you been taught about goodness of fit. What tests can you use? BTW: Only one of those data points fall outside the limits of the fit.

When you want to do a fit to something, you need (at least) two variables. In your case, the first (x = independent variable) is the height. The second would be the percentage (or fractional amount) of people of that height. So how would you go about getting that fractional amount. That is, what charts, kind of analysis, etc. will get you what you want? Once you get that, you then need to decide on what to use for a fit. Generally a least squares fit [the kind probably used here] is 'less complicated' and looks 'nicer' for lower number of parameters [straight line, quadratic, cubic] and higher number of parameters fits the data better but can be very complicated.

 

[110699]

What have you been taught about goodness of fit. What tests can you use? BTW: Only one of those data points fall outside the limits of the fit.

I performed a chi-squared analysis, which came out with a p-value of 60, showing that the estimated probabilities of each height from 150 to 198 are not statistically significant in comparison to the observed values. I assume that this is a suitable empirical test to show that the estimated pdf is incorrect.

I began creating my own pdf, using turning point form, coming to a conclusion of:

f(x) = -(x-171)(x-201)/4500 for 171<x<201

It checks out, in that integrating it from 171 to 201 is equal to 1. However, I am not sure if this is the correct answer. The data we obtained is nowhere near normally distributed, so I am unsure whether the calculated pdf is appropriate. Thanks.