Section 9 – 5A:
Testing a Claim about the Mean of the Differences between Matched Pairs Test H 0: µd = 0 (there is no difference in Population Means ...
Testing a Claim about the Mean of the Differences between Matched Pairs Test H 0: µd = 0 (there is no difference in Population Means of the matched pairs) against ud > 0
or
H1: ud < 0
or
H1: µd ≠ 0
at a significance level of α with DF = n − 1 Requirements 1. The sample data consists consists of matched pairs. 2. The samples are simple random samples. 3. The number of matched pairs is greater than 30 or the population of the differences is normal. Notation for the Samples of Two Population Means of Matched Pairs d = the individual differences between each matched pair. n = the number of matched pairs in the sample
µd = the average (mean) of all the differences d in the population of matched pairs. d = the mean( average) of all the differences in the sample of matched pairs. sd = the standard deviation of all the differences in the sample of matched pairs.
Testing a Claim about the Mean of the Differences between Matched Pairs Dependent Samples with H 0: µd = 0 Test Statistic:
Testing a Claim about the Mean of the Differences between Matched Pairs Example 1(Left Tail Test) Eight Folsom Lake college students selected at random were tested for resting heart rates. They were then shown an exciting 30 minute video lecture of Mr. Eitelʼs statistics class. A second reading of their heart rate was taken after the video The table below shows the heart rates for the before and after readings. Use a α = .05 significance level to test the claim that watching the exciting statistics video can lower heart rates. Assume the population of differences is normal. Student
A
B
C
D
E
F
G
H
Before
40
45
67
43
56
72
81
92
After
38
42
65
44
52
72
80
88
Difference = d After – Before
–2
–3
2
–1
–4
0
–1
–4
A general review of the differences would lead you to claim that the heart rate may be lower after the statistics video. H 0: ud = 0 The results from putting the list of differences into the calculator. H1: ud < 0
d = −1.625
α = .05
n=8
Left Tail Test of H 0: ud = 0
Test Statistic: (d ) t= ⎛ sd ⎞ ⎜ ⎟ ⎝ n⎠
DF = 7
Reject H0
α = .05
sd = 2.066
Do Not Reject H0
t= t
t = –1.895
(−1.625) ⎛ 2.066 ⎞ ⎜ ⎟ ⎝ 8 ⎠
t = −2.22 Conclusion based on H 0 :
Reject H0
Conclusion based on the problem: There is sufficient evidence at the α = .05 level to support the claim that watching the exciting statistics video can lower heart rates.
Testing a Claim about the Mean of the Differences between Matched Pairs Example 2 (Right Tail Test) A PE Teacher felt that hearing a recorded pep talk from Hulk Hogan would increase the number of sit ups a group of 12 years olds could do in 5 minutes. A random group of 12 year olds were selected and the number of sit ups they could do was recorded. The next day they listened to a recorded pep talk from Hulk Hogan and a second test was given. Use a α = .01 significance level to test the claim that hearing a recorded pep talk from Hulk Hogan can help increase the number of sits ups a 12 year old can do in 5 minutes. Assume the population of differences is normal. Student
A
B
C
D
E
F
G
H
I
J
Before
23
12
25
32
15
16
21
17
11
14
After
24
13
20
34
16
16
22
12
15
10
1
1
5
2
1
0
1
–5
4
–3
Difference = d After – Before
A general review of the differences would lead you to guess that the number of sit ups may increase after listening to the recording of Hulk Hogan. H 0: ud = 0 The results from putting the list of differences into the calculator. H1: ud > 0
d = .7
α = .01
sd = 2.94
n = 10
Right Tail Test of H 0: ud = 0
Test Statistic: d t= sd n
DF = 9 Do Not Reject H0
Reject H0
α = .01
t= t
.7 2.94 10
t = 2.821 t = .75 Conclusion based on H 0 :
Do not Reject H0
Conclusion based on the problem: There is not sufficient evidence at the α = .01 level to reject the hypothesis that hearing a recorded pep talk from Hulk Hogan will not increase the number of sits ups a 12 year old can do in 5 minutes. or There is not sufficient evidence at the α = .01 level to support the claim that hearing a recorded pep talk from Hulk Hogan can help increase the number of sits ups a 12 year old can do in 5 minutes. Section 9 – 5A Lecture
Testing a Claim about the Mean of the Differences between Matched Pairs Example 3 (Two Tail Test) Folsom Lake College students must take an assessment test before they can attend their first math class. A random group of 9 students were given the Math 100 placement test and their scores were recorded. They were then allowed to retake the exact same test 4 times and the final score was recorded. Use a α = .05 significance level to test the claim that taking the test 4 times will not change the placement score. Assume the population of differences is normal. Student
A
B
C
D
E
F
G
H
I
Before
32
45
33
28
26
27
32
20
21
After
38
49
35
44
32
36
35
25
28
6
4
2
16
6
9
3
5
7
Difference = d After – Before
A general review of the differences would lead you to claim that there is a change in the placement score after taking the test 4 times. H 0: ud = 0 The results from putting the list of differences into the calculator. H1: ud ≠ 0
d = 6.44
α = .05 so α = .025
n=9
Two Tail Test of H 0: ud = 0 DF = 8
Reject H0
α 2 = .025
Test Statistic: (d ) t= ⎛ sd ⎞ ⎜ ⎟ ⎝ n⎠
Reject H0
Do Not
α 2 = .025
Reject H0
t t = –2.306
sd = 4.15
t = 2.306
t=
(6.44 )
⎛ 4.15 ⎞ ⎜ ⎟ ⎝ 9 ⎠
t = 4.66 Conclusion based on H 0 :
Reject H0
Conclusion based on the problem: There is sufficient evidence at the α = .01 level to support the claim that taking the test 4 times will change the placement score.
