Elementary Statistics and Computer Application

Large and Small sample theory

Large sample theory

• The sample size n is greater than 30 (n≥30) it is known as large sample. For large samples the sampling distributions of statistic are normal(Z test). A study of sampling distribution of statistic for large sample is known as large sample theory.

Small sample theory

• If the sample size n ils less than 30 (n<30), it is known as small sample. For small samples the sampling distributions are t, F and χ2 distribution. A study of sampling distributions for small samples is known as small sample theory.

Test of Significance

• The theory of test of significance consists of various test statistic. The theory had been developed under two broad heading
• Test of significance for large sample
• Large sample test or Asymptotic test or Z test(n≥30)
• Test of significance for small samples(n<30)
• Small sample test or Exact test-t, F and χ2.
• It may be noted that small sample tests can be used in case of large samples also.

Large sample test
Large sample test are

1. Sampling from attributes
2. Sampling from variables

Sampling from attributes
There are two types of test for attributes

1. Test for single proportion
2. Test for equality of two proportions

Test for single proportion

• In a sample of large size n, we may examine whether the sample would have come from a population having a specified proportion P=Po. For testing
• We may proceed as follows

1.Null Hypothesis (Ho):

• Ho: The given sample would have come from a population with specified proportion P=Po

2.Alternative Hypothesis(H1)

• H1 : The given sample may not be from a population with specified proportion
• P≠Po (Two Sided)
  P>Po(One sided-right sided)
  P<Po(One sided-left sided)

3.Test statistic



• It follows a standard normal distribution with µ=0 and σ²=1

4.Level of Significance

• The level of significance may be fixed at either 5% or 1%
 
5.Expected vale or critical value

• In case of test statistic Z, the expected value is
 
Ze = 1.96 at 5% level

2.58 at 1% level Two tailed test


Ze = 1.65 at 5% level

2.33 at 1% level One tailed test


Inference

• If the observed value of the test statistic Zo exceeds the table value Ze we reject the Null Hypothesis Ho otherwise accept it.