Question

1. a coffee shop advertises that its small brewed coffee contains 12 ounces of coffee. as the manager, you worry the automatic dispenser is underfilling cups. a test of 50 cups gives a mean of 11.82 ounces.

we want to test
$h_0: \mu = 12$
$h_a: \mu < 12$
where $\mu$ = the true mean amount of coffee dispensed. if the p - value = 0.034 what conclusion would you make?
\\(\circ\\) because the p - value of 0.034 < 0.05, we accept $h_0$. we have convincing evidence the true mean is 12 ounces.
\\(\circ\\) because the p - value of 0.034 < 0.05, we fail to reject $h_0$. we do not have convincing evidence that cups are being underfilled.
\\(\circ\\) because the p - value of 0.034 < 0.05, we reject $h_0$. we do not have convincing evidence that cups are underfilled.
\\(\circ\\) because the p - value of 0.034 < 0.05, we reject $h_0$. we have convincing evidence that cups are being underfilled.

Explanation:

Step1: Recall hypothesis testing rule

Reject \(H_0\) if \(p\)-value < significance level (\(\alpha\), typically 0.05).

Step2: Compare p-value to 0.05

Given \(p\)-value = 0.034 < 0.05, so reject \(H_0\).

Step3: Interpret rejection

Rejecting \(H_0: \mu=12\) supports \(H_a: \mu<12\) (cups are underfilled).

Answer:

Because the P-value of 0.034 < 0.05, we reject \(H_0\). We have convincing evidence that cups are being underfilled.