Download shaped and symmetric. At the 1% level of significance, test the claim

Paired Differences ( Dependent ) Worksheet Answers
1. For a random sample of 20 data pairs, the sample mean of the difference was 𝑑̅ = 2.
The sample standard deviation 𝑠𝑑 = 5. Assume that the distribution of the differences is
mound – shaped and symmetric. At the 1% level of significance, test the claim that the
population mean of the difference is positive.
𝐻0 ∶ 𝜇𝑑 = 0
𝑡=
𝑑̅ √𝑛
𝑠𝑑
=
𝐻1 ∶ 𝜇𝑑 > 0
(2)(√20)
5
=
13.42
5
𝛼 = 0.01
= 2.684
with 𝑑. 𝑓. = 19
𝑃 − 𝑣𝑎𝑙𝑢𝑒 of 𝑡 = 2.684 is between 0.010 and 0.005
Since 𝑃 − 𝑣𝑎𝑙𝑢𝑒 ≤ 0.01, we reject 𝐻0
2. Are America’s top chief executive officers ( CEOs ) really worth all that money ? One way
to answer this question is to look at row B, the annual company percentage increase in
revenue, versus row A, the CEO’s annual percentage salary increase in that same
company. Do the data indicate that the population mean percentage increase in
corporate revenue ( row B ) is different from the population mean percentage increase
in CEO salary? Use 𝛼 = 0.05
B : percentage
increase for company
A : percentage
increase for CEO
24
23
25
18
6
4
21
37
21
25
20
14
-4
19
15
30
𝑩−𝑨
3
-2
5
4
10
-15
6
7
Using a calculator : 𝑑̅ = 2.25 , 𝑠𝑑 = 7.78 , 𝑛 = 8 , 𝑑. 𝑓. = 7
𝐻0 ∶ 𝜇𝑑 = 0
𝑡=
𝑑̅ √𝑛
𝑠𝑑
=
𝐻1 ∶ 𝜇𝑑 ≠ 0
(2.25)(√8)
7.78
=
6.36
7.78
= 0.817
𝛼 = 0.05
( two – tailed )
with 𝑑. 𝑓. = 7
𝑃 − 𝑣𝑎𝑙𝑢𝑒 of 𝑡 = 0.817 is between 0.250 and 0.500
Since 𝑃 − 𝑣𝑎𝑙𝑢𝑒 > 0.05, we fail to reject 𝐻0
3. Do professional golfers play better in their first round? Let row B represent the score in
the fourth ( and final ) round, and let row A represent the score in the first round of a
professional golf tournament. A random sample of finalists in the British Open gave the
following data for their first and last rounds of the tournament. Do the data indicate
that the population mean score on the last round is higher than that on the first round ?
Use a 5% level of significance.
B : Last
A : First
𝑩−𝑨
73
66
7
68
70
-2
73
64
9
71
71
0
71
65
6
72
71
1
Using a calculator : 𝑑̅ = 2 , 𝑠𝑑 = 4.5 , 𝑛 = 9 , 𝑑. 𝑓. = 8
𝐻0 ∶ 𝜇𝑑 = 0
𝑡=
𝑑̅ √𝑛
𝑠𝑑
=
𝐻1 ∶ 𝜇𝑑 > 0
(2)(√9)
4.5
=
6
4.5
= 1.333
𝛼 = 0.05
with 𝑑. 𝑓. = 8
𝑃 − 𝑣𝑎𝑙𝑢𝑒 of 𝑡 = 1.333 is between 0.125 and 0.100
Since 𝑃 − 𝑣𝑎𝑙𝑢𝑒 > 0.05, we fail to reject 𝐻0
68
71
-3
68
71
-3
74
71
3
4. In an effort to determine if rats perform certain tasks more quickly if offered a larger
reward, the following experiment was performed. On day 1, a group of three rats, was
given a reward of one food pellet each time they ran a maze. A second group of three
rats was given a reward of five pellets each time they ran a maze. On day two, the
groups were reversed. The average time in seconds for each rat to run the maze 30
times are show in the following table.
Rat
A
B
C
D
E
F
Time with one food pellet
Time with five food pellets
𝑩−𝑨
3.6
3.0
0.6
4.2
3.7
0.5
2.9
3.0
-0.1
3.1
3.3
-0.2
3.5
2.8
0.7
3.9
3.0
0.9
Do these data indicate that rats receiving larger rewards tend to run the maze in less
time? Use a 5% level of significance
Using a calculator : 𝑑̅ = 0.4 , 𝑠𝑑 = 0.45 , 𝑛 = 6 , 𝑑. 𝑓. = 5
𝐻0 ∶ 𝜇𝑑 = 0
𝑡=
𝑑̅ √𝑛
𝑠𝑑
=
(0.4)(√6)
0.45
𝐻1 ∶ 𝜇𝑑 > 0
=
0.98
0.45
= 2.178
𝛼 = 0.05
with 𝑑. 𝑓. = 5
𝑃 − 𝑣𝑎𝑙𝑢𝑒 of 𝑡 = 2.718 is between 0.025 and 0.010
Since 𝑃 − 𝑣𝑎𝑙𝑢𝑒 ≤ 0.05, we reject 𝐻0