MATH1725 Introduction to Statistics: Worked examples

Worked Example: Lectures 5–6
A sample of size 64 is drawn by simple random sampling from a normal population which has
known variance 4. The sample mean is −0.45. Test the hypothesis H 0 : µ = 0 vs. H 1 : µ ≠ 0 at
the 5% level of significance. Repeat for testing H 0 : µ = 0 vs. H 1 : µ > 0
Answer: Here ¯X ∼ N(µ,σ 2 /n) with σ 2 = 4, n = 64, so σ 2 /n = 0.0625 and ¯X ∼ N(µ,0.0625).
Test statistic is
Z = ¯X − µ
σ/ √ n = ¯X
√ =
¯X
0.0625 0.25
where Z ∼ N(0,1) if H 0 is true.
For α = 0.05 with a two-sided test, z α/2 = 1.96. Critical region is Z < −1.96 and Z > 1.96.
Observed value is z = −0.45/0.25 = −1.8. This does not lie in critical region so accept H 0 .
For α = 0.05 with a one-sided test, z α = 1.645. Critical region is Z < −1.645. Observed value
is z = −1.8 which lies in critical region so reject H 0 .
Worked Example: Lecture 6
The absenteeism rates (in days and parts of days) for nine employees of a large company were
recorded in two consecutive years.
Employee 1 2 3 4 5 6 7 8 9
Year 1 3.0 6.7 11.3 5.0 9.4 15.7 8.0 10.0 9.7
Year 2 2.8 5.1 8.4 5.0 6.2 12.2 10.0 6.8 6.0
Is there any evidence that the average absenteeism rate is different for the two years
Answer: Data paired as same employee studied in each of the two years.
Form difference d i = (year 1) i − (year 2) i . Need to estimate variance σ 2 d .
Test H 0 : µ d = 0 vs. H 1 : µ d ≠ 0. See lecture 6.
Worked Example: Lecture 8
Which phrases i-iv below apply to the sample correlation coefficient r XY
(i) measures linear association between two variables,
(ii) is never negative,
(iii) has positive slope,
(iv) depends on the units of measurement of X and Y .
Answer: i only.
Worked Example: Lecture 8
The tensile strength of a glued joint is related to the glue thickness. A sample of six values gave
the following results:
Glue Thickness (inches) 0.12 0.12 0.13 0.13 0.14 0.14
Tensile Strength (lbs.) 49.8 46.1 46.5 45.8 44.3 45.9
Calculate the sample correlation coefficient r for these data.
Use the fitted least squares regression line to predict the tensile strength of a joint for a glue
thickness of 0.14 inches.
Using scatter-diagrams, sketch the form of regression line expected in the three cases when r
takes the values −1, 0, and +1.
5