Statistical Hypothesis Testing and Significance Tests for Attributes and Variables

1.1 Procedure for testing of hypothesis

step 1: Null Hypothesis. Set up the Null Hypothesis H0.

step 2: Alternative Hypothesis. Set up the Alternative Hypothesis H1. This will enable us to decide whether we have to use a single-tailed(right or left) test or two-tailed test.

step 3: Level of significance. Choose the appropriate level of significance (α) depending on the reliability of the estimates and permissible risk. This is to decided before sample is drawn, i.e., α is fixed in advance.

step 4: Test Statistic (or Test Criterion). Compute the test statistic:

Z = t - E(t)/S.E.(t), under H0

step 5: Conclusion. We compare the computed value of Z in step 4 with the significant value (tabulated value) Zα at the given level of significance, ‘α’.

If | Z | < Zα, i.e., if the calculated value of Z is less than Zα, we can say it is not significant. Therefore null hypothesis is accepted.

If | Z | > Zα, i.e., if the calculated value of Z is greater than Zα, we can say it is significant. Therefore null hypothesis is rejected.

1.2 Sampling of Attributes

There are two main test under the sampling of attributes

1.2.1 Test of Significance for Single Proportion

Formula: Z = X - nP/√nPQ

where

X is number of successes

n is independent trials

P is probability of success for each trial

Q = 1-P

1.2.2 Test of Significance for Difference of Proportions

Formula: Z = p1 - p2/√PQ(1/n1 + 1/n2)

where

p1 & p2 are sample proportions from two populations respectively.

n1 & n2 are size of random samples from two populations respectively.

P = (n1*p1) + (n2*p2)/(n1+n2)

Q = 1-P

1.3 Sampling of Variables

1.3.1 Test of Significance for Single Mean

Formula: Z = (x - µ)/(σ/√n)

where

x is sample mean

µ is population mean

n is size of random sample

σ is SD

1.3.2 Test of Significance for Difference of Mean

Formula: Z = (x1- x2)/√{σ1²/n1 +σ2²/n2} or (x1- x2)/σ√{1/n1 +1/n2}

where

x1 is sample mean from population

x2 is sample mean from another population

n1 is size of random sample from population

n2 is size of random sample from another population

σ is SD of population

1.3.3 Test of Significance for Difference of Standard Deviation

Formula: Z = s1 - s2/√(σ1²/n1 + σ2²/n2)

where s1 and s2 are SD of two independent samples

Will continue this blog with examples for each test…