Stat 104 – Lecture 30
Two Independent Samples
• Do males and females at I.S.U.
spend the same amount of time,
on average, at the Lied
Recreation Athletic Center?
1
Populations
random
selection
2. Male
Inference
1. Female
random
selection
Samples
2
Time (minutes)
1. Females
2. Males
63, 32, 86, 53, 49
73, 39, 56, 45, 67
49, 51, 65, 54, 56
52, 75, 74, 68, 93
77, 41, 87, 72, 53
84, 65, 66, 69, 62
3
1
Stat 104 – Lecture 30
Time (minutes)
Sex=F
Mean
Std Dev
Sex=M
55.87 Mean
69.20
13.527 Std Dev
13.790
n
15 n
15
4
Comment
• This sample of I.S.U. females
spends, on average, 13.33
minutes less time at the Lied
Recreation Athletic Center than
this sample of I.S.U. males.
5
Confidence Interval: μ
1
( y1 − y2 ) ± t
*
− μ2
s12 s22
+
n1 n2
t* from Table T,
df = smaller of (n1 − 1, n2 − 1)
6
2
Stat 104 – Lecture 30
Finding t*
• Use Table T.
• Confidence Level in last row.
• df = smaller of (n1 – 1, n2 – 1) =
14.
7
Table T
df
1
2
3
4
M
2.145
14
Confidence Levels 80%
90%
95%
98%
99%
8
s12 s22
+
=
n1 n2
(13.527)2 + (13.792)2
15
15
= 24.88 = 4.988
9
3
Stat 104 – Lecture 30
Confidence Interval: μ
1
( y1 − y 2 ) ± t*
− μ2
s12 s 22
+
n1 n2
(55.87 − 69.2) ± 2.145(4.988)
− 13.33 ± 10.70
− 24.03 to − 2.63
10
Interpretation
• We are 95% confident that
I.S.U. females spend, on
average, from 2.63 to 24.03
minutes less time at the Lied
Recreation Athletic Center than
I.S.U. males do.
11
Test of Hypothesis: μ
1
− μ2
• Step 1: Set – up.
H 0 : μ1 = μ2 or H 0 : μ1 − μ2 = 0
H A : μ1 ≠ μ2 or H A : μ1 − μ2 ≠ 0
12
4
Stat 104 – Lecture 30
Test of Hypothesis: μ
1
− μ2
• Step 2: Test Criteria.
– Times are normally distributed.
– Population standard deviations not
known.
– t-test statistic.
– α = 0.05
13
Test of Hypothesis: μ
1
− μ2
• Step 3: Sample evidence.
t=
( y1 − y 2 ) − 0
s12 s 22
+
n1 n 2
14
Time (minutes)
Sex=F
Mean
Std Dev
n
Sex=M
55.87 Mean
69.20
13.527 Std Dev
13.790
15 n
15
15
5
Stat 104 – Lecture 30
(13.527)2 + (13.792)2
s12 s22
+
=
n1 n2
15
15
= 24.88 = 4.988
16
Test of Hypothesis: μ
1
− μ2
• Step 3: Sample evidence.
t=
( y1 − y2 ) − 0 = (55.87 − 69.20)
4.988
s12 s22
+
n1 n2
t = −2.672
17
Table T
Two tail probability
0.20
0.10
0.05 0.02 P-value 0.01
df
1
2
3
4
M
14
1.345 1.761 2.145 2.624 2.672 2.718
18
6
Stat 104 – Lecture 30
Test of Hypothesis: μ
1
− μ2
• Step 4: Probability value
–The P-value is between 0.01 and
0.02.
19
Test of Hypothesis: μ
1
− μ2
• Step 5: Results.
– Reject the null hypothesis because
the P-value is smaller than α = 0.05
– The difference in mean times is not
zero. Therefore, on average, females
and males at I.S.U. spend different
amounts of time at the Lied
Recreation Athletic Center.
20
Comment
• This conclusion agrees with the
results of the confidence interval.
• Zero is not contained in the 95%
confidence interval (–24.03 mins to
–2.63 mins), therefore the
difference in population mean
times is not zero.
21
7
Stat 104 – Lecture 30
JMP
• Data in two columns.
–Response variable:
• Numeric – Continuous
–Explanatory variable:
• Character – Nominal
22
JMP Starter
• Basic – Two-Sample t-Test
–Y, Response: Time
–X, Grouping: Sex
23
Oneway Analysis of Time By Sex
100
90
Time
80
70
60
50
40
30
F
M
Sex
Means and Std Deviations
Level
F
M
Number
15
15
Mean
55.8667
69.2000
Std Dev
13.5270
13.7903
Std Err Mean
3.4927
3.5606
Lower 95%
48.376
61.563
Upper 95%
63.358
76.837
t Test
F-M
Assuming unequal variances
Difference
-13.333 t Ratio
Std Err Dif
4.988 DF
Upper CL Dif
-3.116 Prob > |t|
Lower CL Dif
-23.550 Prob > t
Confidence
0.95 Prob < t
-2.67326
27.98961
0.0124
0.9938
-15 -10
0.0062
-5
0
5
10
15
24
8