Identify the value of the TEST STATISTIC used in a hypothesis test of the follow

Question

Identify the value of the TEST STATISTIC used in a hypothesis test of the following claim and sample data:



Claim: “The proportion of symptom-free placebo subjects is less than the proportion of symptom-free subjects who received a new supplement.”



A medical researcher wants to determine if a new supplement is effective at reducing the duration of the common cold. Healthy subjects are intentionally infected with a cold-causing virus, and then given either a placebo or the new supplement one day later. Of the 267 subjects who were given a placebo, 33% were symptom-free a week later. And of the 289 subjects who were given the supplement, 56% were symptom-free after a week. Test the claim at the 0.01 significance level.

Explanation / Answer

Let's denote the placebo group by number 1 and the supplement group by 2.

GIven data:

n1 = 267, p1 = 0.33

n2 = 289, p2 = 0.56

Pooled sample proportion, p = (p1n1 + p2n2)/(n1 + n2) = (0.33*267 + 0.56*289)/(267+289) = 0.45

Standard error, SE = (p*(1-p)*(1/n1 + 1/n2))0.5 = (0.45*(1-0.45)*(1/267 + 1/289))0.5 = 0.042

Test statistic, t = (p2-p1)/SE = (0.56-0.33)/0.042 = 5.476

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