Question

Problem 1:

An investigator in a university wants to know the mean math SAT score among all the applications to the university this year (denoted by μ). A random sample of n = 50 applications are selected. The sample mean is X = 530 and the sample variance is s2 = 8100.

a. Consider the hypothesis testing problem: H0 : μ = 520 against Ha : μ ̸= 520. Use a significance level of α = 0.05.

• Compute the test statistic.

• Obtain the rejection region.

• Write down the decision of whether to reject H0.

b. Consider the hypothesis testing problem: H0 : μ = 550 against Ha : μ ̸= 550. Use a significance level ofα = 0.05.

• Compute the test statistic.

• Compute the approximate p-value. (Hint: you can approximate the t distribution by the standard normal distribution)

• Write down the decision of whether to reject H0.

c. Construct a 95% two-sided confidence interval for μ. Suppose we want to test H0 : μ = 500 againstH0 : μ ̸= 500 using a significance level of α = 0.05. Based on this confidence interval, write down the decision of whether to reject H0.

d. Consider the hypothesis testing problem: H0 : μ ≤ 510 against Ha : μ > 510. Use a significance level ofα = 0.1.

• Compute the test statistic.

• Obtain the rejection region.

• Write down the decision of whether to reject H0.

e. Consider the hypothesis testing problem: H0 : μ ≥ 550 against Ha : μ < 550. Use a significance level ofα = 0.025.

• Compute the test statistic.

• Obtain the rejection region.

• Write down the decision of whether to reject H0.

Answer 1

Part a)

Test Statistic :-


t = 0.7857

Test Criteria :-
Reject null hypothesis if


Result :- Fail to reject null hypothesis

Decision :- Fail to reject H0

Part b)

Test Statistic :-


t = -1.5713

Decision based on P value
P - value = P ( t > 1.5713 ) = 0.1225
Reject null hypothesis if P value <    level of significance
P - value = 0.1225 > 0.05 ,hence we fail to reject null hypothesis
Conclusion :- Fail to reject null hypothesis

Decision :- Fail to reject H0

Part c)

Confidence Interval



Lower Limit =
Lower Limit = 504.4223
Upper Limit =
Upper Limit = 555.5777
95% Confidence interval is ( 504.4223 , 555.5777 )

Decision :- Since   lies in the interval, hence we fail to reject H0

Part d)

Test Statistic :-


t = 1.5713

Test Criteria :-
Reject null hypothesis if


Result :- Reject null hypothesis
Decision :- We reject H0

Part e)

Test Statistic :-


t = -1.5713
Test Criteria :-
Reject null hypothesis if


Result :- Fail to reject null hypothesis
decision :- Fail to reject H0