Page 422 - DMTH404_STATISTICS

Statistics

Notes 31.1 Test of Hypothesis

In order to illustrate the procedure of testing a null hypothesis, let us assume that the life of
electric bulbs of a company is distributed normally with standard deviation of 150 hours and we
want to test the null hypothesis that the mean life of bulbs is 1600 hours against the alternative
hypothesis that the mean life is not 1600 hours.

Assuming that H is true, we can construct a sampling distribution of X , the mean life of bulbs
0
in the sample. If a random sample of 100 bulbs is taken from this population, we know that the
150
distribution of X will be normal with mean m = 1600 hours and standard error, . .S E = = 15
X
10
hours. Further, we know that for a normal distribution
æ X - 1600 ö
P - 2 £ £ 2 = 0.9544
÷
ç
è 15 ø
or P (1600 2 15- ´ £ X £ 1600 2 15 ) = 0.9544
+
´
or P (1570 £ X £ 1630 ) = 0.9544

This result shows that the likelihood of getting a random sample, from the given population,
with mean lying between 1570 and 1630 hours is 95.44% or equivalently, the likelihood of
getting a random sample having its mean either less than 1570 or more than 1630 hours is only
4.56%. Thus, a random sample with its mean lying outside these limits is highly unlikely under
the assumption that null hypothesis is true.
However, if the mean computed from the drawn sample is found to lie outside these limits, it
may imply that either null hypothesis is false or the rare event, with probability = 4.56%, has
occurred.
Thus, if we decide to reject the null hypothesis whenever the computed sample mean falls
outside the above limits, the probability of our decision being wrong is only 4.56% or 0.0456.
Two Types of Errors
The decision of acceptance or rejection of a null hypothesis is made on the basis of a sample from
a population and hence, an element of uncertainty is always involved in making such decisions.
Two types of errors are likely to be committed in the procedure of testing a hypothesis. These
are Type I and Type II errors. Type I error is committed when a true null hypothesis is rejected.
The probability of this error is termed as the Level of Significance of the test and will be denoted
by a. The probability of committing an error is also termed as its size. Note that size of type I
error, i.e., a = 0.0456, in the above example.
Contrary to this, type II error is committed when a false null hypothesis is accepted. The
probability of type II error is denoted by b. To understand the meaning of type II error, we
assume that the true value of m is 1620 instead of the hypothesised value of 1600 hours. If the
standard deviation is same, the value of b is given by P (1570 £ X £ 1630) when m = 1620 or P
æ 1570 - 1620 1630 - 1620 ö
z
ç £ £ ÷ = P(–3.33 £ Z £ 0.67) = 0.4996 + 0.2486 = 0.7482
è 15 15 ø

414 LOVELY PROFESSIONAL UNIVERSITY