Question

Problem 1 A study is conducted to investigate long-term complication rates in patients treated with two different therapies. The sample data is given below. Long-term complications Yes No Therapy 1 92 932 Therapy 2 75 878 Let p1 be the population proportion of patients with long-term complications among those treated with therapy 1. Let p2 be the population proportion of patients with long-term complications among those treated with therapy 2. Can we conclude that the proportions of patients with long-term complications are different under these two therapies? Let’s conduct a hypothesis test using a significance level of 0.01.

a. Write down the null hypothesis. (5 points)

b. Write down the alternative hypothesis. (5 points)

c. What’s the point estimate of p1? (5 points)

d. What’s the point estimate of p2? (5 points)

e. What’s the pooled sample proportion? (5 points) f. Calculate the test statistic. (5 points)

g. Find out the critical value. (5 points)

h. What’s the decision whether to reject the null hypothesis? (5 points)

i. Verify that the condition for large sample size is met. (5 points)

Answer 1

Given that,
sample one, x1 =92, n1 =932, p1= x1/n1=0.099
sample two, x2 =75, n2 =878, p2= x2/n2=0.085
null, Ho: p1 = p2
alternate, H1: p1 != p2
level of significance, α = 0.01
from standard normal table, two tailed z α/2 =2.576
since our test is two-tailed
reject Ho, if zo < -2.576 OR if zo > 2.576
we use test statistic (z) = (p1-p2)/√(p^q^(1/n1+1/n2))
zo =(0.099-0.085)/sqrt((0.092*0.908(1/932+1/878))
zo =0.977
| zo | =0.977
critical value
the value of |z α| at los 0.01% is 2.576
we got |zo| =0.977 & | z α | =2.576
make decision
hence value of |zo | < | z α | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != 0.9765 ) = 0.3288
hence value of p0.01 < 0.3288,here we do not reject Ho
ANSWERS
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a.
null, Ho: p1 = p2
b.
alternate, H1: p1 != p2
c.
point estimate of proportion 1 = p1 =0.099
d.
point estimate of proportion 2 = p2 =0.085
e.
the pooled sample proportion = p = ((p1*n1 +p2*n2)/(n1+n2)) = (0.099*932 +0.085*878)/(932 +878) =0.0922
f.
test statistic: 0.977
g.
critical value: -2.576 , 2.576
h.
decision: do not reject Ho
p-value: 0.3288
i.
we do not have enough evidence to support the claim that the proportions of patients with long-term complications are different under these two therapies.