Problem 2 As we'll discuss later in the course, a two-sample t-test is a common way...

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Problem 2 As we'll discuss later in the course, a two-sample t-test is a common way to compare the pop- ulation mean of two groups; this test involves at distribution. Meanwhile, one-way analysis of variance (ANOVA) can be used to compare the population mean of two or more groups; one-way ANOVA involves an F distribution. In this problem, we'll study the relationship between the t distribution and F distribution. Answer the following three questions. Part A Define Ttv, i.e. at distribution with degrees of freedom v. For this part, derive the following quantities: E[T] E[T] Var(T) Please show your work for how you arrived at your answers. Your final answer for E[T] and Var(T) should involve v. Furthermore, you should find that, technically, your answer for E[T] and Var(T) will only be valid if >2; that is totally fine for the sake of this problem. Part B A common rule-of-thumb associated with the Central Limit Theorem is that it is valid to use if the sample size is approximately n 30. Because of this, when comparing two sample means, sometimes researchers will use a two-sample z test (which uses a standard Normal distribution) instead of a two-sample t test (which uses at distribution). For this part, answer the following two questions. Let's say n = 30. In this case, the t distribution used for the two-sample t test is tn-2 = t28. What is the ratio of the variance for at distribution with 28 degrees of freedom to the variance of a standard Normal distribution? . Let's say T Part C 2 t28 and Z 2 N(0, 1). As we discussed in class, p-values correspond to the probability that a statistic is greater than a particular value, under the assumption that the statistic follows a particular distribution. For this part, use R to compute P(T 1.7) and P(Z 1.7). In your answer, please include the R code you used to compute these probabilities, as well as numeric answers for the probabilities themselves. . Let F 2 For this part, answer the following two questions. Fv1,v2 denote an F distribution with degrees of freedom v and v2. Given this, derive E[F]. Please show your work for how you arrived at your answer, which should involve v and/or v2. (Hint: You do not need to work with a pdf to do this.) As stated in Part B, given a sample size of n, a two-sample t-test involves the tn-2 distribu- tion. Meanwhile, as stated in the beginning of this problem, one-way ANOVA can be used to compare sample means for two or more groups, which involves an F distribution. When there are just two groups, ANOVA involves the F1,n-2 distribution. Thus, let T~ tn-2 and F~ F1,n-2. How does E[T2] compare to E[F]? Problem 2 As we'll discuss later in the course, a two-sample t-test is a common way to compare the pop- ulation mean of two groups; this test involves at distribution. Meanwhile, one-way analysis of variance (ANOVA) can be used to compare the population mean of two or more groups; one-way ANOVA involves an F distribution. In this problem, we'll study the relationship between the t distribution and F distribution. Answer the following three questions. Part A Define Ttv, i.e. at distribution with degrees of freedom v. For this part, derive the following quantities: E[T] E[T] Var(T) Please show your work for how you arrived at your answers. Your final answer for E[T] and Var(T) should involve v. Furthermore, you should find that, technically, your answer for E[T] and Var(T) will only be valid if >2; that is totally fine for the sake of this problem. Part B A common rule-of-thumb associated with the Central Limit Theorem is that it is valid to use if the sample size is approximately n 30. Because of this, when comparing two sample means, sometimes researchers will use a two-sample z test (which uses a standard Normal distribution) instead of a two-sample t test (which uses at distribution). For this part, answer the following two questions. Let's say n = 30. In this case, the t distribution used for the two-sample t test is tn-2 = t28. What is the ratio of the variance for at distribution with 28 degrees of freedom to the variance of a standard Normal distribution? . Let's say T Part C 2 t28 and Z 2 N(0, 1). As we discussed in class, p-values correspond to the probability that a statistic is greater than a particular value, under the assumption that the statistic follows a particular distribution. For this part, use R to compute P(T 1.7) and P(Z 1.7). In your answer, please include the R code you used to compute these probabilities, as well as numeric answers for the probabilities themselves. . Let F 2 For this part, answer the following two questions. Fv1,v2 denote an F distribution with degrees of freedom v and v2. Given this, derive E[F]. Please show your work for how you arrived at your answer, which should involve v and/or v2. (Hint: You do not need to work with a pdf to do this.) As stated in Part B, given a sample size of n, a two-sample t-test involves the tn-2 distribu- tion. Meanwhile, as stated in the beginning of this problem, one-way ANOVA can be used to compare sample means for two or more groups, which involves an F distribution. When there are just two groups, ANOVA involves the F1,n-2 distribution. Thus, let T~ tn-2 and F~ F1,n-2. How does E[T2] compare to E[F]?