Use at least 5 decimals in your calculations in this question. The university would like to see whether the math course of linear algebra can help students improve grades in the econometrics class. They select two groups of students. The students in one group are a random sample of students who took the math course before the econometrics class ( X population). The students in the other group are an independent random sample of students who did not take the math course before the econometrics class ( Y population). Assume student course scores are approximately normally distributed in each population. Assume the population variances are unknown but the same for two. In a random sample of 23 students from the X population (who took the math course), the mean econometrics course scores were 80 and the standard deviation was 8 . In an independent random sample of 16 students from the Y population (who did not take the math course), the mean econometrics course scores were 70 and the standard deviation was 6. 1. Use the rejection region approach to test the null hypothesis that the mean econometrics course scores are the same in the two populations of students, against the alternative hypothesis that the means are different. Use a 10% significance level. Give the rejection region in terms of the test statistic Xˉ − Yˉ. Be sure to include the sampling distribution of the test statistic and the reason for its validity in the problem as part of your answer. 2. Give the 90% confidence interval. Use this confidence interval to reach a conclusion in the hypothesis test about the means of the populations (from the first question). Be sure to explain how you reach a conclusion. 3. Test the null hypothesis that the variances of the distributions of econometrics course scores in the two populations are the same against the alternative hypothesis that the variances are different. Use the rejection region approach and a 10% level of significance. 4. Calculate the 90% confidence interval for σ^2/x/σ^2/y2​. Explain how to use the calculated confidence interval to reach a conclusion in a test of the null hypothesis that the variances of the populations are the same, against the alternative hypothesis that the variances are different, at a 10% level of significance.