# Basic understanding: Confidence interval for discrete variable

Let's say I play a game with where the outcome can be any even number from 10 to 30, with unknown probabilities. After playing this game, I have the following results:

$$\bar X=20, s=3$$ (with $$n=1000$$)

If I were to calculate the 95% confidence interval, using the formula:

$$\bar X \pm 1.96\cdot s/\sqrt n$$

1. Is this even valid given that the underlying distribution is discrete?
2. Disregarding #1, am I correct in assuming that the resulting confidence interval tells me that the true mean will lie inside this range 95% of the time?
3. Similarly, am I correct in assuming that it does not tell me that the outcome value will lie within this interval 95% of the time?
4. How can I find the 95% confidence interval for the outcome values?
• The formula you give is for a confidence interval for a mean, (suitable when $X\sim\,\text{iid}$ normal, though it would often also be suitable in very large i.i.d samples). When you ask for an interval for a variable, you'd have to clarify which of several kinds of interval you seek instead (e.g. a prediction interval? a tolerance interval? perhaps something else?), and such a calculation will depend on the underlying distribution even when you collected a large sample. Nov 3, 2018 at 7:16

• Thank you. In regards to #4, what I meant to ask was: How do I find the values x and y so that I can say: If I repeated this game a large number of times, the outcome (my score) would be between x and y 95% of the time? Nov 2, 2018 at 11:29