A researcher wants to show the mean from population 1 is less than the mean from population 2 in matched-pairs data. if the observations from sample 1 are xi and the observations from sample 2 are yi, and diequalsximinusyi, then the null hypothesis is h0: mudequals0 and the alternative hypothesis is h1: mud ___ 0.

The null hypothesis is represented as

H₀: μd = 0

The alternative hypothesis is represented as

H₁: μd < 0, that is, mud less than 0.

Step-by-step explanation:

Correct Question

A researcher wants to show the mean from population 1 is less than the mean from population 2 in matched-pairs data. if the observations from sample 1 are xi and the observations from sample 2 are yi, and di = (xi -yi), then the null hypothesis is h0: μd = 0 and the alternative hypothesis is h1: μd ___ 0.

Solution

For hypothesis testing, the first thing to define is the null and alternative hypothesis.

In hypothesis testing, especially one comparing two sets of data, the null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test. It usually maintains that, with random chance responsible for the outcome or results of any experimental study/hypothesis testing, its statement is true.

The alternative hypothesis usually confirms the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test. It usually maintains that significant factors other than random chance, affect the outcome or results of the experimental study/hypothesis testing and result in its own statement.

For this question, we are aiming to show that the mean from population 1 (μₓ) is less than the mean from population 2 (μᵧ).

The null hypothesis would be that there isn't significant evidence to suggest that the mean from population 1 (μₓ) is less than the mean from population 2 (μᵧ). That is, the mean from population 1 (μₓ) is not less than the mean from population 2 (μᵧ).

The alternative hypothesis is that there is significant evidence to suggest that mean from population 1 (μₓ) is less than the mean from population 2 (μᵧ).

Mathematically, if μₓ and μᵧ are the means of population 1 and 2 respectively and μd is the mean of the differences between population 1 and population 2, μd = μₓ - μᵧ

The null hypothesis is represented as

H₀: μₓ = μᵧ or μd = 0

The alternative hypothesis is represented as

H₁: μₓ < μᵧ or μd < 0

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