Question

Consider a large-sample level 0.01 test for testing H₀: p = 0.2 against H_a: p > 0.2.

(a) For the alternative value p = 0.21, compute β(0.21) for sample sizes n = 100, 1600, 12,100, 40,000, and 90,000. (Round your answers to four decimal places.)

n	β
100	_____
1600	_____
12100	_____
40000	_____
90000	_____

(b) For p̂ = x/n = 0.21, compute the P-value when n = 100, 1600, 12,100, and 40,000. (Round your answers to four decimal places.)

n	P-value
100
1600
12100
40000

(c) In most situations, would it be reasonable to use a level 0.01 test in conjunction with a sample size of 40,000? Why or why not?

Yes, even when the departure from H₀ is significant from a practical point of view, a statistically significant result is not likely to appear; it is difficult for the test to detect departures from H₀.

Yes, it is always advantageous to have a very large sample size, because it will detect very small departures from H₀.    

No, even when the departure from H₀ is insignificant from a practical point of view, a statistically significant result is highly likely to appear; the test is too likely to detect small departures from H₀.

No, it is never advantageous to have a very large sample size, because it cannot detect very small departures from H₀.

Answer 1

a)

n	beta
100	0.9793
1600	0.9032
12100	0.3336
40000	0.0040
90000	0.0000

b)

n	p value
100.0000	0.4013
1600.0000	0.1587
12100.0000	0.003
40000.0000	0.0000

No, even when the departure from H₀ is insignificant from a practical point of view, a statistically significant result is highly likely to appear; the test is too likely to detect small departures from H₀.