Question

An experiment compared the taste of two types of coffee, a store brand and a brand of coffee called Freshly Brewed. Each randomly selected participant tastes two unmarked cups in random order and states, which he or she prefers. Of the 50 people sampled, 28 preferred Freshly Brewed coffee. Let p denote the true proportion of all people who prefer Freshly Brewed coffee.

a) Calculate the p-value of the test. State your conclusion for the hypothesis: Reject Ho or Do not reject Ho?

b)  Find the power of the test when p is actually 0.6 and a = 0.05.

c) How can we increase the power of the test? State all the methods and briefly justify why.

Answer 1

a)

Ho: p = 0.5

Ha: p 0.5

TS = (p^ - p)/sqrt(pq/n)

X = 28

n = 50

p^ = X/n = 0.56

TS = (0.56 - 0.5)/sqrt(0.5*0.5/50)

= 0.8485

p-value = 2 P(Z > |TS|) {as this is 2-tailed test}

= 2 P(Z > 0.8485)

= 0.3961

since p-value > alpha

we do not reject the null hypothesis

b)

Z = (p^ - p)/sqrt(pq/n)

power = reject the null hypothesis when it is false

= 1 - P( 0.5 - 1.96*sqrt(0.5*0.5/50) < p^ < 0.5 + 1.96*sqrt(0.5*0.5/50) | p = 0.6)

= 1- P( 0.3614 < p^ < 0.63859 | p = 0.6)

= 1- P((0.3614 - 0.6)/sqrt(0.6*0.4/50) < Z < (0.63859 - 0.6)/sqrt(0.6*0.4/50) )

= 1- P ( -3.44389 < Z < 0.556998 )

= 0.2891

c)

we increase power by

i)

increasing sample size

ii) decreasing significance level