Null and Alternative Hypothesis (2012)

Specifying the Null and Alternative Hypothesis of the following Statistical Tests:

1) The t-test for independent samples,
2) One-way analysis of variance,
3) The t-test for correlation coefficients?
4) The t-test for a regression coefficient.
5) Chi-Square Goodness of Fit Test

Before writing the Null and Alternative Hypothesis for each of the above, understand the following in general about the Null and Alternative hypothesis.
In each of these, the null hypothesis says there is no relationship or no difference. The alternative hypothesis says that there is a relationship or there is a difference. The null hypothesis of a test always represents “no effect” or “no relationship” between variables, while the alternative hypothesis states the research prediction of an effect or relationship.

The Null and Alternative hypothesis for each of the above is as follows:

1. In this case, the null hypothesis says that the two population means (i.e., $\mu_1$ and  $\mu_2$) are equal; the alternative hypothesis says that they are not equal.
   
   $H_0: \mu_1 = \mu_2$
   
   $H_1: \mu_1 \ne \mu_2$ or $H_1:\mu_1 > \mu_2$ or $H_1:\mu_1 < \mu_2$
2. In this case, the null hypothesis says that all of the population means are equal; the alternative hypothesis says that at least two of the means are not equal. If there are 4 populations to be compared then
   
   $H_0: \mu_1=\mu_2=\mu_3 = \mu_4$
   
   $H_1:$ at least two population means are different
3. In this case, the null hypothesis says that the population correlation (i.e., $\rho$) is zero; the alternative hypothesis says that it is not equal to zero.
   
   $H_0: \rho = 0$
   
   $H_1: \rho \ne 0$ or $H_1: \rho > 0$ or $H_1: \rho < 0$
4. In this case, the null hypothesis says that the population regression coefficient ($\beta$) is zero, and the alternative says that it is not equal to zero.
   
   $H_0: \beta_1 = 0$
   
   $H_1: \beta_1 \ne 0$
5. In this case, the null hypothesis says that there is no association between categories of Variable-1 and categories of variable-2. The alternative hypothesis says that there is an association between categories of Variable-1 and categories of Variable-2.
   
   $H_0:$ There is no association between grouping variables
   
   $H_1:$ There is an association between grouping variables

MCQs about Number System (Intermediate Mathematics Part-I