Question

use a t - test to test the claim about the population mean \\( \mu \\) at the given level of significance \\( \alpha \\) using the given sample statistics. assume the population is normally distributed.
claim: \\( \mu \geq 7800 ; \alpha=0.05 \\) sample statistics: \\( \bar{x}=7500, s = 430, n = 24 \\)
what are the null and alternative hypotheses?
\\( \bigcirc \\) a. \\( h _ { 0 } : \mu = 7800 \\)
\\( h _ { a } : \mu \
eq 7800 \\)
\\( \bigcirc \\) c. \\( h _ { 0 } : \mu \leq 7800 \\)
\\( h _ { a } : \mu > 7800 \\)
\\( \bigcirc \\) b. \\( h _ { 0 } : \mu \geq 7800 \\)
\\( h _ { a } : \mu < 7800 \\)
\\( \bigcirc \\) d. \\( h _ { 0 } : \mu \
eq 7800 \\)
\\( h _ { a } : \mu = 7800 \\)

Explanation:

In hypothesis testing, the null hypothesis \(H_0\) is a statement of equality or no difference. The alternative hypothesis \(H_a\) is the claim we are trying to find evidence for. The claim is \(\mu\geq7800\). Since the null hypothesis must have an equality (\(=\), \(\leq\), or \(\geq\)) and the alternative hypothesis is the complement. When the claim is \(\mu\geq7800\), the null hypothesis \(H_0:\mu\geq7800\) and the alternative hypothesis \(H_a:\mu < 7800\) (because we test against the complement of the claim when the claim has an equality - like \(\geq\) or \(\leq\)).

Answer:

B. \(H_0:\mu\geq7800\), \(H_a:\mu < 7800\)