A radio station in Myrtle Beach announced that at least 90 %...

A radio station in Myrtle Beach announced that at least $90 \%$ of the hotels and motels would be full for the Memorial Day weekend. The station advised listeners to make reservations in advance if they planned to be in the resort over the weekend. On Saturday night a sample of 58 hotels and motels showed 49 with a no-vacancy sign and 9 with vacancies. What is your reaction to the radio station's claim after seeing the sample evidence? Use $a=.05$ in making the statistical test. What is the $p$-value?

Key Concepts

One-Sample Proportion Hypothesis Test

This test evaluates whether the proportion in a single sample meets a predefined claim about the population proportion. It begins with stated hypotheses—typically the null hypothesis asserts that the true proportion equals or exceeds a claim (for example, at least 90%), and the alternative hypothesis contends that it is less. The method involves calculating the sample proportion and then determining how far that observed value deviates from the claimed proportion in the context of expected sampling variability.

p-value

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one computed from the sample data if the null hypothesis is true. It provides a measure of evidence against the null hypothesis. A smaller p-value indicates stronger evidence for rejecting the null hypothesis, and it is compared to the significance level to make a determination about the hypothesis.

Significance Level (?)

The significance level, often denoted by ?, is the threshold used to decide when to reject the null hypothesis. It represents the maximum tolerated probability of making a Type I error—rejecting a true null hypothesis. In hypothesis tests, if the p-value is less than or equal to ?, the result is considered statistically significant, leading to rejection of the null hypothesis.

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