Question

Independent random samples of n₁ = 15 and n₂ = 12 observations were selected from two normal populations with equal variances.

Population
1 2
Sample Size 15 12
Sample Mean 34.9 32.1
Sample Variance 4.2 5.2
(a) Suppose you wish to detect a difference between the population means. State the null and alternative hypotheses for the test.

a. H₀: (₁ − ₂) = 0 versus Ha: (₁ − ₂) ≠ 0
b. H₀: (₁ − ₂) < 0 versus Ha: (₁ − ₂) > 0
c. H₀: (₁ − ₂) ≠ 0 versus Ha: (₁ − ₂) = 0
d. H₀: (₁ − ₂) = 0 versus Ha: (₁ − ₂) < 0
e. H₀: (₁ − ₂) = 0 versus Ha: (₁ − ₂) > 0

Answer 1

The null and alternative hypotheses for the test is a. H₀: (μ₁ − μ₂) = 0 versus Ha: (μ₁ − μ₂) ≠ 0

How is it so?

Null Hypothesis (H₀): Stated assumption that there is no difference between the population means.

- In mathematical terms, this is expressed as
`\( \mu_1 - \mu_2 = 0 \)`, where
`\( \mu_1 \) and \( \mu_2 \)` are the population means.

- Alternative Hypothesis (Ha): Stated assumption that there is a difference between the population means.

- In this case, the alternative hypothesis is expressed as
`\( \mu_1 - \mu_2 \\eq 0 \)`, indicating a two-tailed test where we are interested in detecting any difference, whether it is positive or negative.

So, the correct pair is
`\[ H_0: \ (\mu_1 - \mu_2) = 0 \ \text{versus} \ H_a: \ (\mu_1 - \mu_2) \\eq 0 \]`

This is option "a" from your choices.