Suppose a random sample of 100 observations from a binomial

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.

Solution

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)

Z_tab = 1.96 at = 0.05 level of significance

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