Suppose you randomly choose a sample of 30 observations from a normal distribution. You then use those observations to estimate the mean of the population.

(a) Assuming that we would consider anything from 158 to 162 as "equal to 160," and that we know that the standard deviation of the population is 1.0, estimate the probability that we will be able to determine that the mean is different than 160 if it actually is different than 160. Use α = 0.05. (Hint: this is equivalent to saying, "what is the probability that we will reject the null hypothesis given that it is false?")

(b) If the mean of the observations is 157.2, what would you guess the mean of the population is (on the basis of the sample)? (I do not want the confidence interval just report your estimate of the population mean.)

(c) Suppose the population is N (160.0, 1.2). It is possible that you could extract a sample from that population such that a hypothesis test on that sample would reject the null hypothesis, even though it is true. If we use α = 0.01, what are the chances of this happening? (Hint: this is equivalent to saying, "what is the probability that we would reject the null hypothesis given that it is true?")

(d) If you were to accept the null hypothesis, even though the null hypothesis was false, what type of error would you have committed?