Question

For this test, do the following.

(a) Identify the claim and state H0 and Ha.

(b) Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test, a t-test, or a chi-square test. Explain your reasoning.

(c) Choose one of the options. If convenient, use technology.

Option 1: Find the critical value(s), identify the rejection region(s), and find the appropriate standardized test statistic.

Option 2: Find the appropriate standardized test statistic and the P-value.

(d) Decide whether to reject or fail to reject the null hypothesis.

(e) Interpret the decision in the context of the original claim.

3. A travel agent says that the mean hotel room rate for a family of 4 in a certain resort town is at most $170. A random sample of 33 hotel room rates for families of 4 has a mean of $179 and a standard deviation of $19. At a = 0.01, is there enough evidence to reject the agent’s claim?

Answer 1

(a) The following null and alternative hypotheses need to be tested:

$H_0:\mu\le170$

$H_1:\mu>170$

(b) This corresponds to a rightt-tailed one-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

(c) Based on the information provided, the significance level is $\alpha = 0.05,$ $df=n-1=32$ degrees of freedom, and the critical value for a right-tailed test is $t_c = 1.693889.$

The rejection region for this right-tailed test is $R = \{t:t>1.693889\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{179-170}{19/\sqrt{33}}\approx2.7211$$

(d) Since it is observed that $t=2.7211>1.693889=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for right-tailed, $df=32$ degrees of freedom, $t=2.7211,$ is $p=0.005218 ,$ and since $p= 0.005218 <0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

(e) Therefore, there is enough evidence to claim that the population mean $\mu$

is greater than 170, at the $\alpha = 0.05$ significance level.