A quality expert inspects 380 items to test whether the population proportion of

Question

A quality expert inspects 380 items to test whether the population proportion of defectives exceeds .04, using a right-tailed test at = .02.

(a)

What is the power of this test if the true proportion of defectives is [formula108.mml] = .05? (Round intermediate calculations to 4 decimal places. Round your answer to 4 decimal places.)

  Power   =?

(b)

What is the power of this test if the true proportion is [formula108.mml] = .06? (Round intermediate calculations to 4 decimal places. Round your answer to 4 decimal places.)

  Power   =?

(c)

What is the power of this test if the true proportion of defectives is [formula108.mml] = .07? (Round intermediate calculations to 4 decimal places. Round your answer to 4 decimal places.)

  Power =?

Explanation / Answer

Standard error of sampling distribution of proportion = sqrt(p(1-p)/n)

SE = sqrt(0.04*(1-0.04)/380) = 0.01

Critical value of z for = .02 is 2.05

Let Critical value of proportion be pc, then

(pc - p) / SE = 2.05

(pc - 0.04) / 0.01 = 2.05

pc = 0.04 + 0.01 * 2.05 = 0.0605

So, if observed proportion is greater than 0.0605, we reject the null hypothesis.

When p = 0.05 (So, null hypothesis is False)

Power of the test = P(rejecting the null hypothesis given that it is not true) = P(the test statistic will fall in the rejection region when the null hypothesis is false)

= P[p > 0.0605 | mean = 0.05] = P[ z > (0.0605 - 0.05) / 0.01] = P(z > 1.05)

= 0.1468

b)

When true proportion is 0.06

Power = P[p > 0.0605 | mean = 0.06] = P[ z > (0.0605 - 0.06) / 0.01] = P(z > 0.05)

= 0.4801

c)

When true proportion is 0.07

Power = P[p > 0.0605 | mean = 0.07] = P[ z > (0.0605 - 0.07) / 0.01] = P(z > -0.95)

= 0.8289

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