Which of the following is an appropriate null hypothesis?
$H_o: \mu_o=\mu_a$
$H_0: \mu_0 \neq \mu_a$
$H_0: \mu_0>\mu_a$
$H_0: \mu_0>\mu_a$
Answer (1)
The null hypothesis ( H 0 ) states there is no effect or no difference.
It always includes an equality sign.
Comparing means μ o and μ a , the null hypothesis is μ o = μ a .
Therefore, the correct null hypothesis is: H o : μ o = μ a
Explanation
Understanding the Null Hypothesis The null hypothesis, denoted as H 0 , is a statement that there is no effect or no difference. It always includes an equality sign. In this case, we are comparing two population means, μ o and μ a . The null hypothesis should state that there is no difference between these means.
Evaluating the Options Let's examine the given options:
H o : μ o = μ a : This states that the mean of population 'o' is equal to the mean of population 'a'. This is a valid null hypothesis because it includes an equality sign and suggests no difference between the means.
H 0 : μ 0 = μ a : This states that the mean of population 'o' is not equal to the mean of population 'a'. This is an alternative hypothesis, not a null hypothesis, because it uses an inequality sign.
\mu_a"> H 0 : μ 0 > μ a : This states that the mean of population 'o' is greater than the mean of population 'a'. This is also an alternative hypothesis because it uses an inequality sign.
\mu_a"> H 0 : μ 0 > μ a : This is the same as option 3 and is also an alternative hypothesis.
Conclusion Therefore, the appropriate null hypothesis is the one that states there is no difference between the means, which is H o : μ o = μ a .
Examples
In medical research, a null hypothesis might state that a new drug has no effect compared to a placebo. For example, H 0 : μ drug = μ placebo would mean the average effect of the drug is the same as the average effect of the placebo. Researchers then conduct experiments to see if there's enough evidence to reject this null hypothesis in favor of an alternative hypothesis, such as the drug having a positive effect.
