Independent t-Test

1
Independent t-Test

• slides from Geoff Browne

2
??? ????? ?? ?????? ???? ???? ??????? ????? ?? ??
??????? ????? ???????
3
?? ??? ?????? ?? ?? ???? ???? ?? ???? ???? ??
???? ?????? ?????? ??????
? ?? ???? ?? ?? ?? ?????? ????
4
.... ????? ???? ?? ?? ???? ???? ?? ????? ????????
???
??? ?
5
???? 1
????? ???????? ??? ?? ???? ???? ????? ?? ???? ??
???? ?????? ???? C1 ?? ???? ?????? ???? D8 ????
??????.
7 ????
11 ????
6
???? 2
?? ??? ????? ?? ?? ???? ???? ????? ?? ?? ???? D8
? ?? ???? ?? ?? ???? C1 ?????? ???? ???.
(9 ????)
(9 ????)
7
????? ?? ????? ?? ????? ???? ???? ????? ?? ????
?????? ???? ???? ?????? ????!!!
8
??? ????? ???? ?????? ?? ??? ?? ???? ???? ?????
???? ....
.... ????? ?? ??????? ?? ?? ???? ???? ???? ....
.... ?????? ????
9
??? ????? ???? ?????? ?? ??? ?? ???? ???? ??????
???? .....
.... ????? ?? ??????? ?? ?? ???? ???? ???? .....
..... ???? ?? ????
10
???????
11
???? ???? ??? ?? ??????? ?? ?? ?? ????? ?? ??????
????? ????
??? ??? ????? ???? ?? ??? ??? ????? ?? ???????
?? ????? ?? ? ?? ?????? ???? ???? ??? ?? ???????
?????
12
?? ?????? ?? ????? ????????? ????? ??? ??????? ??
???? ?????? ?? ????? ????
?? ??? ????????? ????? ?? ????? ???????? ?????
?????? ??? ...
... ????? ??? ??????? ?? ? ?? ????? ????????
??????? ?? ????.
13
?? ??? ????????? ????? ?? ????? ???????? ?????
?????? ??? ...
... ????? ??? ?? ????? ???? ??? ??????? ?????.
????? ?? ??????? ?? ???????? ???? ?? ???? ?????
?????? ?????? ???
14
?? ????? ????? ????? ???? ??? ????? ??? ?? ?????
??????? ??? ?? ???? ???? ??????? ?? ?? ????? ??
?????? ?????? .....
... ? ?????? ???????? ???? ?? ?? ????? ????????
??????.
??????? ???? ?????? ????? ?? ????? ?? ?? ?????
?????? ???????? ????? ?????? ?? ??????.
?????? ????? ?? ??????? ?? ????? ??? ?????? ??
????
15
Students t-test
????? t? ??????? ?? ? ?????? ???????? ?? ????? ??
?? ?? ?????? ?? ????? ?? ???? ??? ??? ??? ????
?????? ???????? ???? ?????
???? ??? ????? ????? ????? ??? H0 ? H1 ???? ????
???? ??? ??? t ?????? ???
??? t ?? ??????? ?? ????? ??? ?????? ?? ???
16
Students t-test
??? ??? t ?????? ??? ?? ???? ?? ???? ????? t
?????? ????.
???? ??? ????? ???? ?? ???? ????? ????? ?? ?? ???
??? ???? ?? ???

• ?????? ??? t ? ?? ???? ?? ?????? ?????? ???? ???
  ??? (a0.05)
• ????? ????? ??? ? ???? ????? ?????? ??? ?????
  ????? ?? ????
• ?? ????? ?? ??? t ?????? ??? ?? ??? t ????
  ?????? ????? ????? ??? ???????
• ?? ???.
• ?????? t ?????? ??? ?? ??? t ???? ?????? ?????
  ????? ??? ????? ????? ?? ???.

17
Worked Example Random samples were taken of
pupils in C1 and D8
Their recorded heights are shown below
Students in C1 Students in C1 Students in C1 Students in C1 Students in C1 Students in D8 Students in D8 Students in D8 Students in D8 Students in D8
Student Height (cm) 145 149 152 153 154 148 153 157 161 162
Student Height (cm) 154 158 160 166 166 162 163 167 172 172
Student Height (cm) 166 167 175 177 182 175 177 183 185 187
Step 1 Hypothesis synthesis -
Step 2 Work out the mean height for each sample
168.27
161.60
Step 3 Work out the difference in means
6.67
18
Step 4 Work out the standard deviation for each
sample
C1 s1
10.86
D8 s2
11.74
Step 5 Calculate s2/n for each sample
10.862 15
7.86
11.742 15
9.19
19
Step 6 Calculate
Step 7 Calculate t
(Step 2 divided by Step 5)
20
Step 8 Work out the number of degrees of freedom
d.f. (n1 n2 ) 2
(15 15) 2
28
Step 9 Find the critical value of t for the
relevant number of degrees of freedom
Use the 95 (p0.05) confidence limit
Critical value 2.048
Our calculated value of t is below the critical
value for 28d.f., therefore, there is no
significant difference between the height of
students in samples from C1 and D8
21
?? ?? ?????? ????? ? ???? ????????? ?? ????? ??
???? ??? ????? ??? ???. ?????? ????? ???? ?????
??????? ???? ????? ?? ?????? ? ????? ?????? ??
????? ??? ?? ????
??????? ???? ????? ?????? ?????? 3.19 ???????
???? ????? ????? ?????? 3.80
??? ????? ????? ??? ????? ????? ??? ?? ??? ??
????? t ??????? ?? ??????.
22
?? ??? ?????? ???? ???? ????????? ?? ????? ??
???? ??? ??? ???? ???? ??? ? ??????? ???? ???? ??
?????? ? ????? ?????? ??? ?? ????? ??? ?? ????
??????? ???? ???? ?????? ?????? 45.38 ???????
???? ???? ????? ?????? 45.52
??? ????? ????? ??? ????? ????? ??? ?? ??? ??
????? t ??????? ?? ??????.
23
?? ??? ?????? ???? ????? ????????? ?? ??????? ???
??? ????? ??? ?? ??? ???? ??? ? ?? ??? ??? ??
????
??????? ???? ????? ????????? ??????? ???? ?????
3.75 ??????? ???? ????? ????????? ??????? ????
????? 3.81 ??????? ???? ????? ?????????
??????? ???? 2.70
??? ????? ????? ??? ????? ????? ??? ?? ??? ??
???? ????? ??????? ?? ????
????? ANOWA
24
????? ANOWA
25
????? ANOWA