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Understanding the concept of sample proportion is crucial when dealing with data from a sample instead of a whole population. In the context of hypothesis testing, the sample proportion helps estimate the true proportion of a particular characteristic in the population. To calculate the sample proportion, we take the number of successes (or what we're measuring) and divide it by the total number of observations in our sample. For example, if we're looking at the proportion of hotels filled to capacity, we use: \[ \hat{p} = \frac{X}{n} \] Where:- \( \hat{p} \) is the sample proportion.- \( X \) is the number of successes (hotels with no vacancies).- \( n \) is the total sample size (number of hotels surveyed).In this scenario, \( X = 49 \) and \( n = 58 \), so the calculation is:\[ \hat{p} = \frac{49}{58} \approx 0.8448 \].This means about 84.48% of the sampled hotels showed no vacancies. By comparing this sample proportion to a claimed population proportion, we can begin to see if the claim is statistically valid.

The null hypothesis is a fundamental element of hypothesis testing. It's a statement that there is no effect or no difference, and it represents a starting point for statistical testing. When testing a claim, like the radio station's assertion that at least 90% of hotels are full, the null hypothesis is set as \( H_0: p \geq 0.9 \). It assumes that the proportion of all hotels that are full is at least 90%. We also have an alternative hypothesis, which in this case is \( H_a: p < 0.9 \). This is what we test against the null. If our sample data provides enough evidence against the null hypothesis, we might reject it in favor of the alternative.The null hypothesis is meant to provide a "default" or "neutral" stance, where any deviation must be proven by the data collected and analyzed.

A p-value is a key concept in hypothesis testing that helps determine the significance of your results. It measures the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. In simpler terms, it tells us how likely our sample data would be if the null hypothesis were accurate.To calculate the p-value, we first need the test statistic, calculated here using the formula for the z-test for proportions:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]Where:- \( \hat{p} \) is the sample proportion (0.8448).- \( p_0 \) is the hypothesized population proportion (0.9).- \( n \) is the number of observations (58).Substituting these values, we find:\[ z \approx -1.7769 \]This z-score correlates to a p-value, which can be found using standard normal distribution tables or a calculator, and is approximately 0.0378 in this context.Since this p-value is less than the significance level \( \alpha = 0.05 \), it suggests that the observed data is statistically significant, leading to the rejection of the null hypothesis.