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Final Exam for E&Sc Math IIA
Spring Semester 2003
• You may only use a pocket calculator and a pen. Everything else is
provided (including scratch paper).
• There are 12 problems worth 10 points each.
• 100 points count as 100%. A score of 50% suffices for passing.
• Do not just give the result, but also the formulas you use to get it (in
particular when you are using the calulator).
• Give enough detail so that a reader can follow what you are doing.
(1) Find the rank and a basis of

0
0
0
the kernel of the following matrix.

2 −2 3 −5
1 −1 1 −1
−3 3 −1 −3
(2) Compute the following determinant.
2 1 0
1 0 1
0 1 −2
−1 2 3
−1
2 3 1
(3) Find all solutions of the following linear system.
2 x1 − 3 x2 + 4 x3 = 0
x1 + 2 x2 − x3 = 4
3 x1 + x2 + x3 = 6
(4) Find the eigenvalues and bases of the eigenspaces of the following matrices.
Which are diagonalizable (and why or why not)?
1 2
0 1
(a)
(b)
0 1
2 1
(5) Determine the numbers x for which the series
∞
X
(−3)n
n=0
n+1
xn converges.
(6) Let A and B be two events with probabilities P (A) = 0.5 and P (B) = 0.7.
(a) Without any additional information on A and B, what are the best upper
and lower bounds for P (A ∪ B̄) ?
(b) Assuming that A and B are independent, find P (A ∪ B̄) and P (Ā ∪ B).
(please turn over)
(7) A fair eight-sided die (with numbers from 1 to 8) is thrown three times. Find
the probabilities of the following events as fractions in lowest terms.
(a) Exactly two of the numbers thrown are ≥ 5.
(b) Some number is thrown at least twice.
(8) Assuming that scores of a certain exam are normally distributed with mean 82
points and standard deviation 23 points, what is the minimal score you need to
achieve to be among the top 20% ?
(9) Suppose you have a large (N > 100) sample taken from a normally distributed
population, with sample mean x̄ and sample standard deviation s. Give expressions (in terms of N , x̄ and s) for the lower and upper limits of the 95%
confidence intervals for the population mean µ and population variance σ 2 .
(10) Consider the following pairs (xi , yi ), i = 1, 2, . . . , 15 of data items.
(−10, 5), (−7, 6), (−7, 5), (−4, 2), (−2, 3),
(−1, −1), (3, 1), (4, −2), (7, 1), (8, −2),
(10, −4), (16, −1), (16, −5), (17, −6), (20, −8)
(a) Produce a scatter plot of these data points.
(b) Compute the sample correlation rxy and interpret the result.
(c) Find the line y = mx + c that fits the data best and add its graph to the
scatter plot.
(11) A machine produces steel balls whose diameter is normally distributed. Eight
of these balls are measured to have diameters (in mm)
8.12, 8.02, 7.98, 8.09, 8.07, 8.03, 8.05, 8.11 .
Find the 95% and 99% confidence intervals for the mean of the diameters of the
balls produced by the machine.
(12) Two groups of students write the same exam. For the first group, consisting of
N1 = 72 students, the mean score was x̄1 = 63 points, with a standard deviation
of s1 = 27 points. For the second group, consisting of N2 = 65 students, the
mean score was x̄2 = 75 points, with a standard deviation of s2 = 22 points. At
a level of significance of 1%, can you conclude that one of the two groups was
better prepared for the exam? (We assume that the exam scores in both groups
are normally distributed.)