Statistical Methods

Unit 5 - Large sample test

5.1.Introduction

Sampling distribution: It is the distribution of values of means all possible random samples of same size taken from a population.
If a sample of size ‘n’ is drawn from a normal population with mean ‘µ’ and standard deviation ‘σ’, then sample mean is also distributed as normal with mean ‘µ’ and standard deviation
Standard Error: It is the Standard deviation of the sampling distribution of sample means.

Central limit theorem: If random samples of n measurements are repeatedly drawn from a population with a finite mean µ and standard deviation σ, then the relative frequency histogram drawn for the (repeated) sample means will tend to be distributed normally. Approximation becomes more valid as n increases.
If a sample of size n is drawn from a normal population with mean µ and standard deviation σ, then sample mean is also distributed as normal with mean µ and standard deviation This proposition holds good even if the population from which the sample is drawn is not normal provided the sample size is large (from central limit theorem,). As x is distributed with mean µ and standard deviation the standard normal variate is given by,

Under the null hypothesis H₀ :µ = µ₀

The test statistic is defined by:

The area under the normal curve between µ - 1.96 σ and µ + 1.96 σ is 0.95 Hence, in the case of standard normal variable which has mean zero and variance 1, the area between - 1.96 to 1.96 will be 0.95. Thus, if the hypothesis is true, Z value computed from the sample will be between - 1.96 to 1.96 with probability of 0.95. On the other hand, if computed value of Z lies outside the range - 1.96 to 1.96, it can be concluded that such a sample would arise with only probability of 0.05, if the null hypothesis was true. In this case it is inferred that Z differs significantly from the Value expected under the hypothesis and hence the hypothesis is rejected, at 5% level of significance.

In the tests involving normal distribution, the set of values of Z outside the range - 1.96 to 1.96 constitutes the region of rejection or critical region (Fig. 1).

In the above discussion 5% level of significance was used. As mentioned earlier any level of significance can be used. If 1% level of significance is used, the region of rejection will be outside the range — 2.58 to 2.58 (Fig.2).