Solution:
Sampling error.
19. You believe that the true mean weight of statistics books is 7.0 lbs. A study of 8 randomly
selected statistics books weights (shown below) was conducted to test this claim. Use a signif
icance level of 0.025 and assume that the weights are normally distributed.
6.4, 7.1, 6.1, 6.9, 5.6, 7, 7.7, 8
(a) (1 point) What type of hypothesis test will you use?
Solution:
Use a one sample mean test with
σ
unknown.
(b) (2 points) What are the test’s requirements?
Solution:
(1) Simple random samples, (2) CLT applies.
(c) (1 point) Are the requirements satisfied?
State how they are satisfied.
Solution:
Yes.
Simple random samples used, and population was normally dis
tributed.
(d) (2 points) What are the hypothesis?
Solution:
H
0
:
μ
= 6
.
0,
H
a
:
μ
6
= 6
.
0
(e) (1 point) What
α
will you use?
Solution:
α
= 0
.
025
(f) (2 points) Conduct the hypothesis test. What is the
p
value?
Solution:
>
x
[ 1 ]
6.4
7.1
6.1
6.9
5.6
7.0
7.7
8.0
>
res
= t . t e s t (x ,
mu = 7 ,
a l t e r n a t i v e
= ”two . sided ”)
>
res
One
Sample
t

t e s t
data :
x
t =

0.5315 ,
df = 7 ,
p

value = 0.6115
a l t e r n a t i v e
hypothesis :
true
mean
i s
not
equal
to
7
95
percent
confidence
i n t e r v a l :
Instructor: Anthony Tanbakuchi
Points earned:
/ 10 points
MAT 167: Statistics, Test II
SOLUTIONS p. 12 of 13
6.182678
7.517322
sample
estimates :
mean
of
x
6.85
The
p
value is 0.612.
Or, if you prefer to do it manually, find the
p
value with the test statistic
t
=
¯
x

μ
0
s/
√
n
,
df
=
n

1
>
x
[ 1 ]
6.4
7.1
6.1
6.9
5.6
7.0
7.7
8.0
>
x . bar = mean(x)
>
x . bar
[ 1 ]
6.85
>
s = sd (x)
>
s
[ 1 ]
0.7982123
>
mu0 = 7
>
n = length (x)
>
n
[ 1 ]
8
>
t = (x . bar

mu0)/( s / sqrt (n ))
>
s i g n i f ( t ,
3)
[ 1 ]

0.532
>
p . val = 2
*
pt ( t ,
df = n

1)
>
s i g n i f (p . val ,
3)
[ 1 ]
0.612
The
p
value is 0.612.
(g) (1 point) What is your formal decision?
Solution:
Since
p
val
α
, fail to reject
H
0
.
(h) (2 points) State your final conclusion in words.
Solution:
The sample data does not contradict the claim that statistics text book
weight is equal to 7.0 lbs.
(i) (1 point) If we reject
H
0
, what is the
actual
probability of a Type I error for this study
data?
Solution:
The
p
value = 0.612.
Instructor: Anthony Tanbakuchi
Points earned:
/ 4 points
MAT 167: Statistics, Test II
SOLUTIONS p. 13 of 13
************
End of exam. Reference sheets follow.
************
Instructor: Anthony Tanbakuchi
Points earned:
/
??
points
Statistics Quick Reference
Card & R Commands
by Anthony Tanbakuchi. Version 1.8.2
ANTHONY
@
TANBAKUCHI
·
COM
Get R at:
R commands:
bold typewriter text
1
Misc R
To make a vector / store data:
x=c(x1, x2, ...)
Help: general
RSiteSearch("Search Phrase")
Help: function
?functionName
Get column of data from table:
tableName$columnName
List all variables:
ls()
Delete all variables:
rm(list=ls())
√
x
=
sqrt(x)
(1)
x
n
=
x
∧
n
(2)
n
=
length(x)
(3)
T
=
table(x)
(4)
2
Descriptive Statistics
2.1
N
UMERICAL
Let
x=c(x1, x2, x3, ...)
total
=
n
∑
i
=
1
x
i
=
sum(x)
(5)
min
=
min(x)
(6)
max
=
max(x)
(7)
six number summary :
summary(x)
(8)
μ
=
∑
x
i
N
=
mean(x)
(9)
¯
x
=
∑
x
i
n
=
mean(x)
(10)
˜
x
=
P
50
=
median(x)
(11)
σ
=
∑
(
x
i

μ
)
2
N
(12)
s
=
∑
(
x
i

¯
x
)
2
n

1
=
sd(x)
(13)
CV
=
σ
μ
=
s
¯
x
(14)
2.2
R
ELATIVE STANDING
z
=
x

μ
σ
=
x

¯
x
s
(15)
Percentiles:
P
k
=
x
i
,
(sorted
x
)
k
=
i

0
.
5
n
·
100%
(16)
To find
x
i
given
P
k
,
i
is:
1.
L
= (
k
/
100%
)
n
2. if
L
is an integer:
i
=
L
+
0
.
5;
otherwise i=L and round up.