Page 247 - DMGT209_QUANTITATIVE_TECHNIQUES_II

Quantitative Techniques-II

Notes strength (and sometimes the direction) of the relationship. Each has its use, and they are best
when used together.
There are two types of tests:

 Small Sample Tests
 Large Sample Test

13.1 Small Sample Tests

13.1.1 T-test

T-test is used in the following circumstances: When the sample size n<30.

Example: A certain pesticide is packed into bags by a machine. Random samples of 10
bags are drawn and their contents are found as follows: 50,49,52,44,45,48,46,45,49,45. Confirm
whether the average packaging can be taken to be 50 kgs.
In this text, the sample size is less than 30. Standard deviations are not known using this test. We
can find out if there is any significant difference between the two means i.e. whether the two
population means are equal.

The Student’s T-distribution

Let X , X ...... X be n independent random variables from a normal population with mean m and
1 2 n
standard deviation s (unknown).

 1 2 
When s is not known, it is estimated by s, the sample standard deviation  s    X  X   .
i
 n 1 

X  
In such a case we would like to know the exact distribution of the statistic and the answer
s/ n
to this is provided by t-distribution.
X  
W.S. Gosset defined t statistic as t  which follows t - distribution with (n–1) degrees of
s/ n
freedom.

Features of t-distribution

2
1. Like c - distribution, t-distribution also has one parameter n = n–1, where n denotes
sample size. Hence, this distribution is known if n is known.

2. Mean of the random variable t is zero and standard deviation is , for n > 2.
  2
3. The probability curve of t-distribution is symmetrical about the ordinate at t = 0. Like a
normal variable, the t variable can take any value from– to .
4. The distribution approaches normal distribution as the number of degrees of freedom
become large.

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