Question

Consider the hypothesis test Ho: 41 = 2 against Hui uz with known variances 01 = 9 and o2 = 5. Suppose that sample sizes nj = 11 and 12 = 14 and that X1 = 4.8 and I2 = 8.0. Use a = 0.05. (a) Test the hypothesis and find the P-value. (b) What is the power of the test in part (a) for a true difference in means of 3? (c) Assuming equal sample sizes, what sample size should be used to obtain B = 0.05 if the true difference in means is 3? Assume that a = 0.05. (a) The null hypothesis is not rejected. The P-value is 0.145 Round your answer to three decimal places (e.g. 98.765). (b) The power is 0.74 Round your answer to two decimal places (e.g.98.76). (c) n = n2 = 127 Round your answer up to the nearest integer. Statistical Tables and Charts

Answer 1

a)

		pop 1	pop 2
sample mean x =		4.80	8.00
std deviation σ=		9.000	5.000
sample size n=		11	14
std error σx1-x2=√(σ21/n1+σ22/n2)    =			3.025
test stat z =(x1-x2-Δo)/σx1-x2     =			-1.06
p value : =			0.2891	(from excel:2*normsdist(-1.06)

p value is 0.289

b)

for 0.05 level and two tailed test critival value Zα=					1.9600
rejection region : x1-x2 <=Δ+z*se or x1-x2 >=Δ+z*se or x1-x2 <=-5.9285 or x1-x2>=5.9285

P(Type II error) =P(-5.9285<x1-x2<5.9285 |Δ=3)=P((-5.9285-3)/3.0248<z<(5.9285-3)/3.0248)=P(-2.95<z<0.97)=0.8324

P(Power) =1-type II error =1-0.8324=0.17

c)

Hypothesized mean difference Δo       =			0
true mean difference =Δ =			3
first sample std deviation =σ1   =			9.000
2ndsample std deviation=σ2 =			5.000
for 0.05 level and left tail critical Zα=			1.96
for 0.05 level of type II error critival Zβ=			1.645
n=(Zα/2+Zβ)2(σ12+σ22)/(Δo-Δ)2 =			154