Question

Suppose that ? is the true value of the proportion of a population and you are testing ?0:?=0.5 vs ?1:?≠0.5.
(a) Using ?=0.01, what is the power of the test for ?=100, assuming that ?=0.65?
(b) Using ?=0.01, what sample size is required if we want a power of 0.95, assuming that ?=0.65?

please show the calculations

Answer 1

(a)

Standard deviation of the proportion is:

Test is two tailed. Since so critical values are -2.575 and 2.575. The left critical value of sample proportion is

The right critical value of sample proportion is

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Now the standard deviation of the proportion is:

The left z-score such that and p= 0.65 is

The right z-score such that and p= 0.65 is

The power of the test is:

(b)

We have

The effect size is

For we have

and for 95% power we have

So the requried sample size is

Hence, n = 198