3
.99
2
.95
.90
1
.75
.50
Normal Quantile Plot
For a random sample of 30 male Egyptian skulls from 4000 BCE, measurements on the maximal
breadth (mm) were made by Egyptian archaeologists. We wish to test to see if the mean maximal
breadth for all Egyptian males in 4000 BCE is less than 135 mm. Graphical and numerical
summaries are given below.
0
.25
-1
100.0%
75.0%
50.0%
25.0%
0.0%
.10
.05
-2
.01
-3
12
8
6
Count
10
maximum
141.00
quartile
135.00
median
131.00
quartile
127.75
minimum
123.00
Mean
131.5
Std Dev 4.8405614
Std Err Mean 0.8837616
upper 95% Mean 133.3075
lower 95% Mean 129.6925
N
30
4
2
120
125
130
135
140
145
Skull Breadth (mm)
a) Answer the questions Who? and What? for these data.
Who? Male Egyptian skulls.
What? Maximal breadth of skull (mm).
b) Set up appropriate null and alternative hypotheses.
population parameter you are testing.
Be sure to clearly define the
H 0 : μ = 135 mm
H A : μ < 135 mm
μ is the mean maximal breadth of skull for all male Egyptians in 4000 BCE.
1
c) Verify the nearly normal condition is met. Be sure to comment on all three plots.
• Histogram
The histogram is symmetric and mounded in the middle.
•
Box plot
The box plot is symmetric.
•
Normal quantile plot
The points on the normal quantile plot follow the diagonal line representing
a normal model.
All of these comments are consistent with a sample from a distribution that follows a
normal model. Therefore the nearly normal condition is satisfied.
d) What are the value of the test statistic and the P-value?
t=
(131.5 − 135) =
⎛ 4.84 ⎞
⎜
⎟
⎝ 30 ⎠
− 3.5
= −3.96
0.884
One tailed probability
0.10
0.05
0.025
0.01
0.005
P-value
1.311
1.699
2.045
2.462
2.756
3.96
df
29
The P-value is less than 0.005.
e) Reach a decision using the P-value.
Because the P-value is so small, reject the null hypothesis.
f) State a conclusion in the context of the problem.
The mean maximal breadth of skull for all male Egyptians in 4000 BCE is less than
135 mm.
2