1.If x$\displaystyle \geq$1 is the critical region for testing $\displaystyle \theta=2$ against $\displaystyle \theta=1$ on the basis of single observations from the population $\displaystyle f(x,\theta)=\theta e^{-\theta};0<x<\infty$

Obtain the value of type 1 and type 2 errors.

2.A machine puts out 16 imperfect articles in a sample of 500.After the machine is overhauled, it puts out 3 imperfect articles in a batch of 100. Has the machine improved?

3.A correlation coefficient of 0.72 is obtained from the sample of 29 pairs of observations.Can the sample be regarded as drawn from a bivariate normal population in which true correlation coefficient is 0.80?