Analytical Exercises: 1) Suppose that the p-value for a null hypothesis is 0.023. Which of the following is true? Give a brief justification for your answer. a) You reject the null hypothesis at the 1% level, but not at the 5% level. b) You reject the null hypothesis at both the 1% and 5% levels. c) You fail to reject the null hypothesis at the 1% level, but reject it at the 5% level. d) You fail to reject the null hypothesis at both the 1% and 5% levels. 2) Suppose we have a simple linear regression model: = 1 + 2 + Using a sample of size = 40 observations, we obtain the OLS estimates ̂ 2 = −4.5 and its associated standard error, . . ( ̂ 2) = 2.1. We want to test the null hypothesis 0: 2 = 0 against the alternative hypothesis 1: 2 ≠ 0. a) What is the value of the t statistic for this test? b) What are the degrees of freedom for this test? c) What is the 5% critical value for this test? What is the 1% critical value for this test? d) Can we reject the null hypothesis at the 5% level? What about at the 1% level? e) What is a type I error? What is the probability of committing a Type I error for the test you performed in part d with a 5% significant level? 3) Suppose a researcher wants to investigate the relationship between class size and test score. She obtains data on class size ( ) and test score ( ) from 30 middle schoolers and estimates the following model: = 1 + 2 + 2 And obtains: ̂ 1 = 525.3, . . ( ̂ 1) = 25.2, ̂ 2 = −2.35, . . ( ̂ 2) = 1.23 a) Interpret the estimate ̂ 2. b) Suppose you want to test the null hypothesis that class size has no effect on test score. Write down a null and a two-sided alternative hypothesis for this test. c) Do you reject the null hypothesis from part b at the 5% significance level? d) Construct a 95% confidence interval for 2. e) Construct a 99% confidence interval for 1. f) Suppose you still want to test the null hypothesis that class size has no effect on test score but you now want to use a one-sided alternative hypothesis. Motivate and conduct this test at the 5% significant level. Does your conclusion differ with that from part c? 4) You are told that the 95% confidence interval for a regression parameter has a lower bound of 0.5 and an upper bound of 1.5. Can you reject the null hypothesis that this parameter is zero at the 5% significance level?