QUESTION IN GUJARATI

[scorpiomanoj]

Hi David,

While I was trying to solve Gujarati's problems, I encountered the following question.( Pl excuse me....as I am sending this from my work place pc, I dont remember the exact page in Gujarati text.)
A random variable X follows a normal distribution with mean 200 and variance 100. What is the probability that the random variable is 180 at a significance level of 5%.
My understanding is that the answer should be Zero.

As I don't have Gujarati's solution manual, I am not sure whether Im correct.
Seeking your assistance.

Thanks
Regards
Manoj Kumar Halan.

 

[David Harper CFA FRM]

Hi Manoj,

You may have transposed the question b/c Gujarati would not write the question quite that way for a couple of reasons. He could ask:

P [ X <= 180] which is NORMDIST(180,200,10,true) = 2.28%; i.e., Z = -2 and the CDF is about 2.28%

or what is the 1-5% = 95% confidence interval, which is
200 - (10)(1.95) < X < 200 + (10)(1.95)

so that lower limit will be slightly higher than 180, so 180 is in the "rejection region" - i.e., we are 95% confident with two-tails that 180 will not occur.

maybe that could be restated (but this is still awkward b/c we are not talking about sample mean): if we observe 180, are we 95% confident the mean is 200? No, we reject.

Strictly speaking, the question is hard to answer. Maybe best to get the correct phrasing. Although, it is instructive to see the problem (aside from the minor issue that a normal PDF cannot equal a value, it must be <, or within an interval. Can we note that a normal variable cannot equal "=" a value exactly): basically in this inference, we can either (i) select a confidence and draw the interval to find the lower limit point [i.e., confidence interval approach] or (ii) we can select the lower limit point and compute the implied confidence [i.e., the significance test or the p-value]. The question seems to be doing both at the same time ?!

I hope this helps, funny how we can learn even from mistaken questions...David