# How do we use plug-in distributions to make probability statements?

I am trying to understand the following problem.

Given a normal distribution fitted by a maximum likelihood with $$\mu_{MLE}=1.688$$ and $$\hat{\sigma}_{MLE}=0.1032$$
What is the probability a person from our population has a height between 1.6m and 1.8m?
which is estimated by
$$p(1.6 < X < 1.8 | \mu_{MLE}=1.688,\hat{\sigma}_{MLE}=0.1032) \approx 0.664$$

How do I proceed with this estimation?
I am familiar with calculating a standard error to look up probability via Z-tables, but can't see how it applies here, given we don't know the sample size.

• Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. Mar 10 at 16:32
• And please check your post for typos. From your numbers I get a probability of approximately zero:  diff(pnorm(c(1.8, 1.6), 0.1032, 0.1032))  0  Mar 10 at 16:35
• @Kjetil It looks safe to take $\mu_{\text{MLE}} = 1.688$ from which I surmise it was intended that $\mu_{\text{MLE}} = 1.7,$ the midpoint of the interval from $1.6$ to $1.8.$
– whuber
Mar 10 at 16:46
• Thank you both. I have lost access to the question but have corrected it as @whuber suggests. Mar 10 at 20:52

With the estimated parameters you do get indeed

 diff(pnorm(c(1.6,  1.8), 1.688, 0.1032))
 0.6641901


that approximate probability. But I understand that you want an estimate of this probability taking into account the uncertainty in the mle estimates, as you say

I am familiar with calculating a standard error to look up probability via Z-tables, but can't see how it applies here, given we don't know the sample size.

which is entirely correct. Without knowing the sample size, either directly or indirectly (if you were given a confidence interval for $$\mu$$, say, you could calculate $$n$$ from that and the standard deviation. Knowing $$n$$, you could use simulation, maybe parametric bootstrap, to analyze the uncertainty in the probability estimate. Without that not much can be done. If this is indeed what you intended, a similar post is here: How to estimate $P(x\le0)$ from $n$ samples of $x$?

• I do not see where in the question an uncertainty (e.g., standard error) is needed--although that would be nice, of course. The question asks only to estimate this probability. Since the parameter estimates are MLEs, the MLE of this probability is given by the calculation you provide at the outset and no further work is needed. See stats.stackexchange.com/a/492281/919.
– whuber
Mar 10 at 21:33
• Thank you. Maybe I miscopied the question. I will look out for it again. Mar 10 at 23:16