Suppose a random sample of 100 observations from a binomial population gives a v

Question

Suppose a random sample of 100 observations from a binomial population gives a value of pˆ = .63 and you wish to test the null hypothesis that the population parameter p is equal to .70 against the alternative hypothesis that p is less than .70.

a) nothing that p = .63, what does your intuition tell you? does the value of p appear to contradict the null hypothesis?

b) use the large-sample z-test to test H0: p = .70 against the alternative hypothesis, Ha: p < 70. Use a = .05. How do the test results compare with your intuitive decision from part A?

c) find and interpret the observed significance level of the test you conducted in part B.

Explanation / Answer

Given n =100   p = 0.63 q = 1-0.63=0.37 P = 0.70 Q = 1- P = 1- 0.70 = 0.30

The null hypothesis is

H0 : P = 0.70 i.e., the population parameter is equal to 0.70

Against the alternative hypothesis

H1 : P < 0.70 i.e., the population parameter is less then 0.70(left tailed alternative)

The test statistic

Z = (p – P)/PQ/n N(0,1)

Z = (0.63 – 0.70)/((0.70)(0.30)/100 ) N(0,1)

Z = -0.07/0.0458 N(0,1)

Z = 1.5275 N(0,1)

Ztab = 1.96 at = 0.05 level of significance

Therefore Zcal < Ztab we accept the null hypothesis i.e, the population parameter is equal to 0.70

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