# Standard Error of Correlated Measurements

### Problem specification

There is a product and a measurement process:

• Product ~ $N(\mu_P, \sigma_P)$
• Measurement ~ $N(\mu_M, \sigma_M)$ (the measurement error is independent of the product under investigation)

I wish to characterize the product, i.e. estimate its mean. To this end, $n_P$ products are measured $n_M$ times. Please note that each of the $1, ..., n_P$ products is measured $n_M$ times. Therefore the measurements are not independent but correlated.

### Question

Regarding the SEM, I wish to obtain an expression of the form $$\sigma_{\bar{z}}=\sigma_{\bar{z}}(n_P, n_M; \sigma_P, \sigma_M)=\textrm{your input}$$ to then make a qualified decision on how to properly chose $n_P$ and $n_M$ to satisfy certain accuracy requirements.

### Proposed solution

Let $Z$ denote the random variable of the mean, $X$ the random variable of product and $Y$ the random variable of the measurement. This gives $$Z(n_P, n_M) = \frac{1}{n_P n_M}\sum_{i=1}^{n_P}\sum_{j=1}^{n_M} X_i + Y_{i, j}.$$ Comuting the variance of $Z$ gives $$Var(Z) = \frac{1}{(n_P n_M)^2}\sum_{i=1}^{n_P} Var(n_M X_i) + \sum_{j=1}^{n_M} Var(Y_{i, j}) \\ = \frac{1}{(n_P n_M)^2}\sum_{i=1}^{n_P} n_M^2 Var(X_i) + n_m Var(Y_{i, j})$$ and after simplifying, since $Var(X_i) = Var(X) = \sigma_P^2$ and $Var(Y_{i,j}) = Var(Y) = \sigma_M^2$, we obtain $$\sigma_{\bar{z}}^2 = Var(Z) = \frac{1}{n_P}\left(\sigma_P^2 + \frac{\sigma_M^2}{n_M}\right).$$

Concerning the confidence interval: $$\bar{z} \pm t_{df;1-\alpha/2} \times s_{\bar{z}}$$ with $df = n_P (1+n_M) -1$.

Are these approaches correct?