Let's say I have an array of values drawn from a normal distribution with mean 50 and standard deviation 10. Using python:
d = np.random.normal(50,10,1000)
I take a single random sample of size n = 10 from this distribution:
s = np.random.choice(d, 10)
What process do I go through to get the best estimate of the population mean from the sample, and an estimation of the margin of error?
Obviously I know the population mean and standard deviation in this case, but let's pretend I don't.
I could also take many samples and compute the sampling distribution of the mean, but let's say I can't do that either.
So I just have this single sample. What process do I go through and can I estimate how often my estimate of the population mean will be wrong?
Best Answer
I'm assuming infinite population size.
The best estimate of the population mean is the sample mean, $\bar{x}$.
The best estimate of the population standard deviation is the sample standard deviation, $s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}$
Since the sample size is less than 30 (10 in this case) and the population standard deviation is unknown, I would prefer to use the T-distribution to develop an interval.
$\bar{x} \pm t(\frac{s}{\sqrt{n}})$
where $t$ is the critical value from the $t_{n-1}$ distribution. In your case with n=10 and a desired confidence level of 95%, $t_{9}=1.833$.
More details can be found in this Example.