9) Suppose you want to test the following hypotheses: H0: p 0.25 vs. H1: p > 0.2
ID: 3356076 • Letter: 9
Question
9) Suppose you want to test the following hypotheses: H0: p 0.25 vs. H1: p > 0.25. A random sample of 1200 observations was taken from the population. Answer the following questions and show your Excel calculation for each question clearly:
a) Let p be the sample proportion. What is the standard error of sample proportion (i.e., _p ) if H0 is true?
b) If the sample proportion obtained were 0.275 (i.e., p =0.275), what is its p-value?
c) If your decision rule is to “reject H0 if p > 0.265 and not to reject otherwise”, what is the probability of Type I error?
d) Suppose that the population proportion is known to be 0.27 (i.e., H0 is false). What is the probability of Type II error if the decision rule given in (c) is used?
e) What is the decision rule if you were to test the above hypothesis at = 5% significance level? Specify the cutoff value and the decision rule for accepting/rejecting H0.
f) Instead of doing a right-tailed test specified above, suppose that you want to do a two-tailed test (i.e., H0: p = 0.25 vs. H1: p 0.25) at = 1% significance level. Specify the decision rule that can be used to reject or accept H0 (i.e., the cutoff values for accepting/rejecting H0).
Explanation / Answer
9) Suppose you want to test the following hypotheses: H0: p 0.25 vs. H1: p > 0.25. A random sample of 1200 observations was taken from the population. Answer the following questions and show your Excel calculation for each question clearly:
a) Let p be the sample proportion. What is the standard error of sample proportion (i.e., _p ) if H0 is true?
Standard error of the sample proportion = sqrt(p0*(1-p0)/N) = sqrt [0.25 * 0.75/1200] = 0.0125
b) If the sample proportion obtained were 0.275 (i.e., p =0.275), what is its p-value?
Here Z = (0.275 - 0.25)/ 0.0125 = 2
p - value= 1- Pr(Z >2) = 1 - 0.97725 = 0.02275
c) If your decision rule is to “reject H0 if p > 0.265 and not to reject otherwise”, what is the probability of Type I error?
Here Z = (0.265 - 0.25)/ 0.0125 = 1.2
probability of type I error = Pr(Z > 1.2) = 1 - Pr(Z < 1.2) = 1 - 0.885 = 0.115
d) Suppose that the population proportion is known to be 0.27 (i.e., H0 is false). What is the probability of Type II error if the decision rule given in (c) is used?
ANswer : Standard error of proportion with true populaton mean = sqrt(0.27 * 0.73/1200) = 0.0128
If the decision rule given in (c) is used
Pr(Type II error ) = Pr(p^ < 0.265 ; 0.27; 0.0128)
Z = (0.265 - 0.27)/ 0.0128 = -0.39
Pr(Type II error ) = Pr(Z < -0.39) = 0.3483
e) What is the decision rule if you were to test the above hypothesis at = 5% significance level? Specify the cutoff value and the decision rule for accepting/rejecting H0.
Answer : Here alpha = 5%
The cutiff value = p0 +- Z95% se0 = 0.25 + 1.645 * 0.0125 = 0.27056
Here, decision rule is p^ > 0.27056 so we will reject the null hypothesis.
f) Instead of doing a right-tailed test specified above, suppose that you want to do a two-tailed test (i.e., H0: p = 0.25 vs. H1: p 0.25) at = 1% significance level. Specify the decision rule that can be used to reject or accept H0 (i.e., the cutoff values for accepting/rejecting H0)
Answer : NOw it is a two tailed test. The confidence interval = p^ +- Z95% se0
= 0.25 +- 1.96 * 0.0125
= (0.2255, 0.2745)
so here cutoff values are 0.2255 and 0.2745
so we shall reject the null hypothesis if p^ < 0.2255 and p^ > 0.2745.
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