# Confidence interval for the difference between $X_1$ and $X_2$ or $X_3$

The question:

Find a 99% confidence interval for the difference in mean oxygen consumption of an algae that is in 100% concentrated seawater with algae that is not in 100% seawater (i.e., 50% or 75%).

Is this the correct approach?

If we let $X_1$ be the random variable corresponding to the mean oxygen consumption for algae in 100% seawater, $X_2$ for 50% and $X_3$ for 75% then I want to find the standard error of $X_1+\frac{X_2+X_3}{2}$.

If so, I know how to proceed from there. Otherwise, I don't know how to start this.

• What do the data look like? – whuber May 18 '13 at 21:18
• Welcome to the site. I took the liberty of editing your question to make it more readable. Please make sure it still says what you want it to. In particular, I changed "water" to algae in the question, which I think may have been a typo. – gung May 18 '13 at 21:39
• @whuber we just have 8 numerical data for 100% and 4 entries for 50% and 75% each. The question is the second part of a question about ANOVA analysis if that helps. I'm just not sure how to make one confidence interval for two differences. or is it asking for multiple confidence intervals? – Ben May 20 '13 at 9:20

In the interest of there being a response on this question:

It looks to me like it's asking for a single interval of the type you were thinking of when you wrote the attempted solution:

If we let $X_1$ be the random variable corresponding to the mean oxygen consumption for algae in 100% seawater, $X_2$ for 50% and $X_3$ for 75% then I want to find the standard error of $X_1+\frac{X_2+X_3}{2}$.

However, you have a sign error in that expression at the end there; the "+" after $X_1$ should be "-". The question asks for:

confidence interval for the difference in mean oxygen consumption

so you need the standard error of an estimate of the contrast $\mu_1-\frac{\mu_2+\mu_3}{2}$ (i.e. difference), which you'd form from the relevant sample means. Personally I'd denote those by $\bar X_i$ rather than $X_i$, to emphasize that we're dealing with group means.

Otherwise, you seem to have basically the right sense of the problem.