MATH1725 Introduction to Statistics: Worked examples

Answer: Let X denote the glue thickness and Y the joint strength.
x y x 2 y 2 xy
0.12 49.8 0.0144 2480.04 5.976
0.12 46.1 0.0144 2125.21 5.532
0.13 46.5 0.0169 2162.25 6.045
0.13 45.8 0.0169 2097.64 5.954
0.14 44.3 0.0196 1962.49 6.202
0.14 45.9 0.0196 2106.81 6.426
Totals 0.78 278.4 0.1018 12934.44 36.135
¯x = 0.78
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= 0.131, ȳ = 278.4
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= 46.41, s 2 X = 1 5 {0.1018 − 6(0.131)2 } = 0.00008,
s 2 Y = 1 5 {12934.44 − 6(46.41)2 } = 3.336,
s XY = 1 {36.135 − 6(0.131)(46.41)} = −0.0114.
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Regression line:
r XY = s XY
s X s Y
=
−0.0114
√ 0.00008 × 3.336
= −0.698.
y = ȳ + (x − ¯x) s XY
s 2 X
= 46.4 + (x − 0.13) −0.0114
0.00008
= 64.925 − 142.5x.
At x = 1.4 ′′ this gives y = 44.975 lbs..
Scatter-plots:
r = −1: data lies on a straight line with negative slope.
r = +1: data lies on a straight line with positive slope.
r = 0: data randomly scattered (X and Y independent) or could show case with X and Y having
a non-linear dependence as in the lecture notes. You could even show both of these cases!
Worked Example: Lecture 11
A coin is tossed three times. Let X denote the number of heads and Y the length of the longest
run of heads or tails. Thus HTT gives X = 1 and Y = 2, THT gives X = 1 and Y = 1.
(a) Obtain the joint probabilities of X and Y .
(b) Obtain the marginal probability distribution of X and Y .
(c) If X = 1, what is the distribution of Y
Answer: (a and b) All eight outcomes are equally likely, so occur with probability 1/8.
Outcome HHH HHT HTH HTT THH THT TTH TTT
X 3 2 2 1 2 1 1 0
Y 3 2 1 2 2 1 2 3
Probability 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
Y
1 2 3 p X (x)
0 0 0 1/8 1/8
X 1 1/8 1/4 0 3/8
2 1/8 1/4 0 3/8
3 0 0 1/8 1/8
p Y (y) 1/4 1/2 1/4 Total = 1
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