A professor claims that students whose native language is not English score dierently in the Microeconomics class than students whose rst language is English.

The professor knows that the expected score of students whose rst language is English is 6.42 (out of a maximum of 10 points) with a population standard deviation of 1.163. What he doesn't know is that these numbers are exactly the same for non-native speaker students. Suppose that the exam scores are normally distributed. (The very few values outside the [0, 10] range may be set as 0 and 10, respectively.)

The professor wants to investigate his claim with an experiment. He will pick 20 students at random whose rst language is not English and compare their score to the parameters of native speaker students with a t-test.

a) What is the null hypothesis? What is the alternative hypothesis?

b) Simulate the experiment in R. Set seed to 5. Generate a population of 10000 students. Draw 3000 samples and store the p-values from the t-tests in a matrix.

c) What is the probability that the professor rejects the null hypothesis? Should he? Which error can occur in this scenario: type I or II?