Question

an insurance agent says the standard deviation of the total hospital charges for patients involved in a crash in which the vehicle struck a construction barricade is less than $3800. a random sample of 20 total hospital charges for patients involved in this type of crash has a standard deviation of $4200. at \\(\alpha = 0.05\\) can you support the agents claim? use the p - value method to test the claim.

identify the null and alternative hypotheses. let \\(\sigma\\) be the population standard deviation.

\\(\bigcirc\\) a. \\(h_0: \sigma < \\$3800\\)
\\(h_a: \sigma \geq \\$3800\\)

\\(\bigcirc\\) b. \\(h_0: \sigma > \\$3800\\)
\\(h_a: \sigma \leq \\$3800\\)

\\(\bigcirc\\) c. \\(h_0: \sigma \leq \\$3800\\)
\\(h_a: \sigma > \\$3800\\)

\\(\bigcirc\\) d. \\(h_0: \sigma \geq \\$3800\\)
\\(h_a: \sigma < \\$3800\\)

Explanation:

1. The agent's claim is that the standard deviation (\(\sigma\)) is less than \$3800. In hypothesis testing, the alternative hypothesis (\(H_a\)) represents the claim we are trying to find evidence for. So \(H_a: \sigma < \$3800\).
2. The null hypothesis (\(H_0\)) is the complement of the alternative hypothesis when considering equality. So if \(H_a: \sigma < \$3800\), then \(H_0\) should be \( \sigma \geq \$3800\) (since it includes the equality case and the opposite of the alternative hypothesis' inequality). This matches option D.

Answer:

D. \( H_0: \sigma \geq \$3800 \)
\( H_a: \sigma < \$3800 \)