The manager of the Danvers-Hilton Resort Hotel stated that...

The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is $$\$ 600$$ or less. A member of the hotel's accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of weekend guest bills to test the manager's claim. a. Which form of the hypotheses should be used to test the manager's claim? Explain. $$ \begin{array}{lll} H_0: \mu \geq 600 & H_0: \mu \leq 600 & H_0: \mu=600 \\ H_{\mathrm{a}}: \mu<600 & H_{\mathrm{a}}: \mu>600 & H_{\mathrm{a}}: \mu \neq 600 \end{array} $$ b. What conclusion is appropriate when $H_0$ cannot be rejected? c. What conclusion is appropriate when $H_0$ can be rejected?

The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is $$\$ 600$$ or less. A member of the hotel's accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of weekend guest bills to test the manager's claim.
a. Which form of the hypotheses should be used to test the manager's claim? Explain.
$$
\begin{array}{lll}
H_0: \mu \geq 600 & H_0: \mu \leq 600 & H_0: \mu=600 \\
H_{\mathrm{a}}: \mu<600 & H_{\mathrm{a}}: \mu>600 & H_{\mathrm{a}}: \mu \neq 600
\end{array}
$$
b. What conclusion is appropriate when $H_0$ cannot be rejected?
c. What conclusion is appropriate when $H_0$ can be rejected?

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the claim being tested is split into two complementary statements. The null hypothesis (H?) represents the status quo or a statement of no effect (often expressed as an equality), while the alternative hypothesis (H?) embodies the claim that is being challenged. In this context, even if the manager’s claim is directional, for purposes of conducting the test, the null hypothesis is often stated with equality (e.g., ? = 600) and the alternative hypothesis reflects the direction of the concern (e.g., ? > 600) to capture any significant increase.

One-Tailed Test

A one-tailed (or one-sided) test is used when the research question or claim involves a specific direction of effect. In this scenario, since the concern is that the mean guest bill might be greater than $600, the test is set up to detect an increase beyond 600. The alternative hypothesis is thus one-sided, focusing on whether the mean exceeds the stated value rather than simply being different from it.

Interpreting Hypothesis Test Outcomes

The interpretation of results from a hypothesis test depends on whether the evidence is strong enough to reject the null hypothesis. If H? cannot be rejected, it implies that there is insufficient evidence to support the claim that the mean guest bill exceeds $600. Conversely, if H? is rejected, it provides statistical evidence that the mean guest bill is indeed greater than $600. It is important to note that failing to reject H? does not confirm that H? is true, but rather indicates that the data did not show a statistically significant deviation in the direction specified by H?.

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