independency

I work with survey data and am interested in comparing the responses of different racial/ethnic groups. So for example, do the responses of White, Asian, Black, and Latinx differ significantly on question #1 in my survey. Race/ethnicity, however, is a "select all that apply" question. So, the...

Let X_i\sim N(\mu,\sigma^2) ; where [i=1,2,\ldots,n] Z_i\sim N(0,1) ; where [i=1,2,\ldots,n] Proof that \frac{(\bar X-\mu)}{\sigma} and \sum_{i=1}^n\frac{(X_i-\bar X)^2}{\sigma^2} are independent, which implies \bar X and \sum_{i=1}^n(X_i-\bar X)^2 are independent. If i show that \bar...

Setup: There are two coins in an urn: A fair coin and a biased coin that lands Heads with probability q. At time 0, we choose a coin at random, and then write down unconditional probabilities P(H1), P(H2), ...P(H100). Here, P(Hi) = probability (estimated at time 0) of getting Heads on flip i...