Z score and Test statistic

Question

According to a recent marketing campaign, 120 drinkers of either Diet Coke or Diet Pepsi participated in a blind taste test to see which of the drinks was their favorite. In one Pepsi television commercial, an anouncer states that "in recent blind taste tests, more than one half of the surveyed preferred Diet Pepsi over Diet Coke." Suppose that out of those 120, 47 preferred Diet Pepsi. Test the hypothesis, using α=0.01 that more than half of all participants will select Diet Pepsi in a blind taste test by giving the following: 

(a)    the test statistic     

(b)    the critical z score

Answer 1

H_o: p_o ≤ 0.5

H_a: p_o > 0.5

p-hat = X/n = 47/120 = 0.39

The sample proportion (p-hat) is those drinkers prefer Diet Pepsi.

We will go ahead and use the one sample proportion test to find the z-statistic and the critical z-value.

z = p₀ - (p-hat) / √(p₀(1 - p₀) /n = 0.5 - 0.39 / √(0.39*0.61)/120 = 0.11/0.045 = 2.44

By using the z-crtical table, the crtical value should be ±2.28 since the level of significance is 0.01.

Since 2.44 > 2.28, we can reject the null hypothesis. There is sufficient evidence to support that more than half the drinkers prefer Diet Pepsi over Diet Coke.