Question

Consider the following competing hypotheses and accompanying sample data. (You may find it useful to reference the appropriate table: z table or t table) H0: p1 − p2 ≥ 0 HA: p1 − p2 < 0 x1 = 250 x2 = 275 n1 = 400 n2 = 400 a. Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.) b. Find the p-value. p-value < 0.01 0.01 ≤ p-value < 0.025 0.025 ≤ p-value < 0.05 0.05 ≤ p-value < 0.10 p-value ≥ 0.10 c. At the 5% significance level, what is the conclusion to the test? Reject H0 since the p-value is less than significance level. Reject H0 since the p-value is greater than significance level. Do not reject H0 since the p-value is less than significance level. Do not reject H0 since the p-value is greater than significance level. d. Interpret the results at α = 0.05. We conclude that the population proportions differ. We cannot conclude that the population proportions differ. We conclude that population proportion 2 is greater than population proportion 1. We cannot conclude that population proportion 2 is greater than population proportion 1.

Answer 1

Given that,
sample one, x1 =250, n1 =400, p1= x1/n1=0.625
sample two, x2 =275, n2 =400, p2= x2/n2=0.688
null, Ho: p1 >= p2
alternate, H1: p1 < p2
level of significance, α = 0.05
from standard normal table,left tailed z α/2 =1.64
since our test is left-tailed
reject Ho, if zo < -1.64
we use test statistic (z) = (p1-p2)/√(p^q^(1/n1+1/n2))
zo =(0.625-0.688)/sqrt((0.656*0.344(1/400+1/400))
zo =-1.861
| zo | =1.861
critical value
the value of |z α| at los 0.05% is 1.64
we got |zo| =1.861 & | z α | =1.64
make decision
hence value of | zo | > | z α| and here we reject Ho
p-value: left tail - Ha : ( p < -1.861 ) = 0.03137
hence value of p0.05 > 0.03137,here we reject Ho
ANSWERS
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null, Ho: p1 >= p2
alternate, H1: p1 < p2
test statistic: -1.861
critical value: -1.64
decision: reject Ho
p-value: 0.03137
Reject H0 since the p-value is less than significance level
we have enough evidence to support the claim that population proportion 2 is greater than population proportion 1