Question

a politician claims that the mean salary for managers in his state is more than the national mean, $83,000. assume the the population is normally distributed and the population standard deviation is $7700. the salaries (in dollars) for a random sample of 30 managers in the state are listed. at \\( \alpha = 0.07 \\), is there enough evidence to support the claim? use technology.
78,945 83,509 89,466 79,248 98,556 80,394 76,746 75,022 88,173 81,659
77,026 87,817 83,225 97,502 85,025 82,836 81,053 91,161 71,290 81,179
72,479 86,600 81,846 75,618 89,265 70,529 88,004 87,083 78,496 98,930
(a) identify the null hypothesis and alternative hypothesis.
a. \\( h _ { 0 } : \mu \leq 83,000 \\)
\\( h _ { a } : \mu > 83,000 \\)
b. \\( h _ { 0 } : \mu \
eq 83,000 \\)
\\( h _ { a } : \mu = 83,000 \\)
c. \\( h _ { 0 } : \mu > 83,000 \\)
\\( h _ { a } : \mu \leq 83,000 \\)
d. \\( h _ { 0 } : \mu = 83,000 \\)
\\( h _ { a } : \mu \
eq 83,000 \\)
e. \\( h _ { 0 } : \mu > 83,000 \\)
\\( h _ { a } : \mu \leq 83,000 \\)
f. \\( h _ { 0 } : \mu \geq 83,000 \\)
\\( h _ { a } : \mu < 83,000 \\)

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 politician claims that the mean salary \(\mu\) is more than \(83000\). So the null hypothesis should be the opposite (non - claim) of the alternative. If the claim is \(\mu>83000\), then the null hypothesis \(H_0:\mu\leq83000\) (since the null and alternative must be mutually exclusive and cover all possibilities for the parameter \(\mu\)).

Answer:

A. \(H_0:\mu\leq83,000\), \(H_a:\mu > 83,000\)