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