Let Y be a Bernoulli random variable with success probability Pr ( Y = 1 ) = p , and let Y 1 , … , Y n be i. i.d. draws from this distribution. Let ˆ p be the fraction of successes (1s) in this sample. Show that ˆ p = ¯¯¯ Y . Show that ˆ p is an unbiased estimator of p. Show that var ( ˆ p ) = p ( 1 − p ) / n .

A. find the power of the test, when the null hypothesis assumes a population mean of mu = 450, with a population standard deviation of 156, the sample size is 5 and the true mean is 638.47 with confidence intervals of 95 b. find the power of the test, when the null hypothesis assumes a population mean of mu = 644, with a population standard deviation of 174, the sample size is 3 and the true mean is 744.04 with confidence intervals of 95% c. find the power of the test, when the null hypothesis assumes a population mean of mu = 510, with a population standard deviation of 126, the sample size is 10 and the true mean is 529.44 with confidence intervals of 95