Pascal is a high-level programming language used frequently in microprocessors. An experiment was conducted to investigate the proportion of Pascal variables that are array variables (in contrast to scalar variables, which are less efficient in terms of execution time). Twenty variables are randomly selected from a set of Pascal programs and Y, the number of array variables, is recorded. Suppose we want to test the hypothesis that Pascal is a more efficient language than Algol, in which 20% of the variables are array variables. That is, we will test H_o: p = .20 against H_a: p > .20, where p is the probability of observing an array variable on each trial. (Assume that the 20 trials are independent.)
a. Find α for the rejection region Y ≥ 8.
b. Find α for the rejection region Y ≥ 5.
c. Find β for the rejection region Y ≥ 8 if p = .5.
d. Find β for the rejection region Y ≥ 5. if p = .5.
e. Which of the rejection regions, Y ≥ 8 or Y ≥ 5, is more desirable if you want to minimize the probability of a Type I error? Type II error?
f. Find the rejection region of the form Y ≥ α so that a is approximately equal to .01.
g. For the rejection region determined in part f, find the power of the test, if in fact p = .4.
h. For the rejection region determined in part f, find the power of the test, if in fact p = .7.