Page 431 - DMTH404_STATISTICS

Unit 31: Hypothesis Testing

We have to test H :  =  against H :    . Notes
0 X Y a X Y
Since sample sizes are large (> 30), it is a large sample case.

36.5 36.8 0.3
-
The test statistic is z = = = 1.219
cal 2 2
1.8 1.5 0.246
+
100 80
Since this value is less than 2.58, there is no evidence against H at 1% level of significance and
0
thus, the observed difference between average life times cannot be regarded as significant.

Example 17: Measurements performed on random samples of two kinds of cigarettes
yielded the following results on their nicotine content (in mgs)
Brand A : 21.4, 23.6, 24.8, 22.4, 26.3
Brand B : 22.4, 27.7, 23.5, 29.1, 25.8

Assuming that the nicotine content is distributed normally, test the hypothesis that brand B has
a higher nicotine content than brand A.
Solution.

We have to test H :  ³  against H :  <  .
0 A B a A B
Note that the rejection of H would imply that brand B has a higher nicotine content than
0
brand A.
The means of the two samples are

21.4 23.6 24.8 22.4 26.3
+
+
+
+
X = = 23.7
A
5
+
+
+
+
22.4 27.7 23.5 29.1 25.8
and X = = 25.7.
B
5
2 2
Also å (X - X A ) = 14.96 and å (X - X B ) = 31.30
Bi
Ai
14.96 31.30
+
The pooled estimate of s is s = = 2.40
+
-
5 5 2
(23.7 25.7- ) 5 5
´
Thus, the test statistic is t = ´ = - 1.318.
cal
2.40 5 5
+
The critical value of t at 5% level of significance and 8 d.f. is - 1.86. Since t is greater than this
cal
value, it lies in the region of acceptance and hence, there is no evidence against at 5% level of
significance. Thus, the nicotine content in brand B is not higher than in brand A.
Example 18: Two salesmen A and B are working in a certain district. From a sample
survey conducted by the head office, the following results were obtained. State whether there is
any significant difference in the average sales between the two salesmen:

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