QUIZ 9: STAB57H3 - An Introduction to Statistics FIRST NAME: LAST NAME: STUDENT NUMBER: TUTORIAL: Problem 1: The director of admissions of a small college selected 5 students at random from the new freshman class in a study to determine whether a student’s grade point average (GPA) at the end of the freshman year came from a N (µ, σ 2 ) with unknown mean µ and variance σ 2 . The GPA of the five students are given below: 3.897 3.885 3.778 1.860 2.948 Given x¯ = 3.27, and s2 = 0.78, compute standardized residuals for the five students. [4 Points] Solution: The standardized residuals are computed as: r n ∗ (xi − x¯) ri = 2 s (n − 1) Hence, the standard residuals are: i: xi : ri∗ : 1 2 3.897 3.885 0.794 0.779 3 3.778 0.643 1 4 1.860 -1.785 5 2.948 -0.408 Problem 2: Calculate normal scores for the five students. Plot the normal scores against the standard residuals. Do you think that the director’s assumption is reasonable? [2 + 2 + 2 = 6 Points] Solution: The order statistics are given as: (i): 1 2 x(i) : 1.860 2.948 3 3.778 4 3.885 5 3.897 The normal score corresponding to the i order statistic is computed using Φ−1 (i/(n + 1)). Hence, the normal scores are: (i): 1 x(i) : 1.860 Score(i) : -0.967 2 3 4 5 2.948 3.778 3.885 3.897 -0.431 0.000 0.431 0.967 1.0 0.5 ● 0.0 ● −0.5 ● ● −1.0 Normal Scores 1.5 The normal probability plot is given below: −2.0 −1.5 ● −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 Standardized Residuals Figure 1: Normal scores against standardized residuals. The points lie reasonably around a straight line with intercept 0 and slope 1. The assumption that the data came from a normal distribution seems reasonable. (Since the sample size is small, students might interpret the results differently.) 2 Appendix 1. Φ−1 (0.1666667) = −0.9674216. 2. Φ−1 (0.3333333) = −0.4307273. 3. Φ−1 (0.5000000) = 0.0000000. 4. Φ−1 (0.6666667) = 0.4307273. 5. Φ−1 (0.8333333) = 0.9674216. 3

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