Question

Explain why the null hypothesis Upper H 0 : mu 1 equals mu 2 is equivalent to the null hypothesis Upper H 0 : mu 1 minus mu 2 equals 0. Choose the correct answer below. A. They are equivalent through algebraic manipulation. B. They are equivalent because the null​ hypothesis, Upper H 0​, is always assumed to be true. C. By​definition, the null hypothesis is always equal to 0.​ Therefore, these hypotheses are equivalent. D. The values of mu 1 and mu 2 are equivalent in every population.​ Therefore, these hypotheses are equivalent.

Answer 1

The null hypotheses 'Upper H 0 : mu 1 equals mu 2' and 'Upper H 0 : mu 1 minus mu 2 equals 0' mean the same thing, representing no difference between two population means, and their equivalence is the result of algebraic manipulation.

Step-by-step explanation:

The null hypothesis Upper H 0 : mu 1 equals mu 2 asserts that there is no difference between the two population means. The alternate hypothesis can be presented as H1: mu 1 does not equals mu 2, which suggests that there is a difference between the population means.

When we state Upper H 0 : mu 1 minus mu 2 equals 0, we are expressing the same concept that there is no difference between the two means, since subtracting two equal values gives a result of zero. This is just another way of saying that the means are the same. The equality of these two hypotheses is a result of algebraic manipulation. Therefore, the correct choice is A: They are equivalent through algebraic manipulation.

Answer 2

The null hypothesis H0: μ1 = μ2 is equivalent to the null hypothesis H0: μ1 – μ2 = 0 because they represent the same idea of ​​no difference between the means of two populations.

Explanation:

The null hypothesis H₀: μ1 = μ2 is equivalent to the null hypothesis H₀: μ1 – μ2 = 0 because it represents the same idea that there is no difference between the means of the two populations. If we subtract μ2 ​​​​from both sides of the first null hypothesis, we get the second null hypothesis. This is because subtracting the values ​​on both sides of the equation does not change the equality ratio.