Quiz Solutions: Hypothesis Testing, t-Stats, Standard Errors in Probability - Prof. Ramiro, Quizzes of Mathematics

Download Quiz Solutions: Hypothesis Testing, t-Stats, Standard Errors in Probability - Prof. Ramiro and more Quizzes Mathematics in PDF only on Docsity! Probability and Statistics Name: Solutions Quiz #7 1) A researcher test the null hypothesis H0 : x = 10 vs. Ha : x > 10; where x is the sample mean. Is this a valid test if signi…cance? In either case, justify your answer. Tests of signi…cance are about the population mean, not the sample mean. 2) De…ne the following: a) The standard error of the sample mean: SEx = spn b) the one-sample t-statistic: t = x SEx c) 2-sample t-statistic: t = x1x2r s21 n1 + s22 n2 d) (extra +1) the pooled estimator: s2p = (n11)s21+(n21)s22 n1+n22 3) What assumption are we making about the population standard deviation when using t-statistics? That the population standard deviation is unknown. 4) What is the main enemy of t-distributions, i.e. when don’t we use t-distributions? When there are outliers, or when the sample is strongly non-normal. 5) Given that 2P [T  4:23] = :005 with df = 6; …nd P [T  4:23] ; where T has a tdistribution (Hint: drawing picture could be helpful) 2P [T  4:23] = :005) P [T  4:23] = :005 2 = 0:002 5: Since T distributions are symmetric, it follows that P [T  4:23] = 0:002 5: 6)a) If n = 6 then df = , where df =degrees of freedom: df = 5 b) (4) =?; where is the gamma function we discussed in class: (4) = 3! = 6: 7) Let x = x1 x2; where x1 = X11+X21+:::+Xn1n1 , x2 = X12+X22+:::+Xn2 n2 ; and x1 and x2 are independent. Find the standard deviation of x and show how you got it. 2x =  2 x1x2 =  2 x1 + 2x2 = 21 n1 + 22 n2 ) x = q 21 n1 + 22 n2 : 8) a) Draw a tdistribution overlapping a normal distribution in the same graph. Indicate which is which. See notes or text. b) What happens as the degrees of freedom in a t distribution get bigger? It approaches a standard normal distribution. 1