We look at the probability of making a Type 1 error in Hypothesis testing. In the graph at left, we put the situation in context and in the graph at right, we put the situation in "standard, normalized" form.

See if you understand the definition of e1L and e2L as integrals. See if you understand the definition of xR and xL (normalized). Change n (size of sample). What can you say about the probability of making a type 1 error? Change xbarR (the non-rejection region). What can you say about the probability of making a type 1 error? Here is the R code: #Givens mi=50 n=10 sig=2.5 #Assume Ho should not be rejected if xbar is between xbarL and xbarR xbarL=48.5 xbarR=51.5 #Find probability of type 1 error. stderr=sig/sqrt(n) xL=(xbarL-mi)/stderr xR=(xbarR-mi)/stderr err1L=pnorm(xL) err1R=1-pnorm(xR) err1=err1L+err1R err1 #err1 is like alpha, i.e. 1-err1 is the probability that you #correctly NOT rejected the Ho. #Notice that we do not calculate type 1 error if we are given alpha. #Assume 1-tailed, e.g. we should not reject Ho if xbar<xbarR. #Eliminate all "L" code in the above.