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The z-test for the mean is a statistical method used to determine if there is a significant difference between the sample mean and a known population mean. It requires the population standard deviation to be known, making it applicable in scenarios where this information is available.

The formula for the z-test is: \[ z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \]where:

• \( \bar{x} \) is the sample mean,
• \( \mu \) is the population mean,
• \( \sigma \) is the population standard deviation,
• \( n \) is the sample size.

In the provided exercise, we used a z-test because we know the population standard deviation \( \sigma = 20,700 \). By using this formula, we calculated the test statistic \( z \approx 2.6 \). This value tells us how many standard deviations the sample mean is away from the population mean. The larger the z-value, the larger the difference between the sample mean and the population mean.

The p-value is a crucial part of hypothesis testing. It represents the probability of obtaining a test statistic at least as extreme as the one calculated if the null hypothesis is true. A smaller p-value indicates that the observed data is unlikely under the null hypothesis. In other words, it helps quantify the evidence against the null hypothesis.

For example, with a one-tailed z-test and a calculated \( z \)-value of \( 2.6 \), we find the p-value using a standard normal distribution table or calculator. In this case, the p-value is approximately 0.0047.

This p-value is quite small, which suggests that the observed sample mean is significantly higher than the expected mean under the null hypothesis. When making decisions, especially in hypothesis testing, understanding the p-value helps determine whether or not to reject the null hypothesis.

The significance level, denoted by \( \alpha \), is a threshold set prior to conducting a hypothesis test. It indicates how willing we are to make a Type I error, which is rejecting a true null hypothesis. Common significance levels used in hypothesis testing include 0.05, 0.01, and 0.10, with smaller values indicating stricter criteria for rejection.

In our exercise, a significance level of \( \alpha = 0.01 \) was chosen, indicating a stringent requirement for rejecting the null hypothesis. A significance level helps determine the critical value or zone of rejection for a test statistic. If the p-value is less than or equal to the significance level, the null hypothesis is rejected.

In our case, the p-value of 0.0047 is less than 0.01, which means we reject the null hypothesis. By setting a low significance level, the analysis aims to minimize errors, requiring strong evidence against the null hypothesis before concluding against it.

In hypothesis testing, the null and alternative hypotheses form the foundation of any analysis. The null hypothesis, denoted as \( H_0 \), is a statement that there is no effect or no difference; it serves as a default assumption. Conversely, the alternative hypothesis, denoted as \( H_a \), posits that there is an effect or a difference.

For this exercise, the null hypothesis \( H_0 \) states that the mean income of Bay Area theatergoers is equal to or less than the general mean for all Playbill readers, \( \mu \leq 119,155 \). Meanwhile, the alternative hypothesis \( H_a \) suggests that the mean income is greater than \( 119,155 \).

The process of hypothesis testing allows us to use sample data to decide whether to reject the null hypothesis. It involves comparing the observed data against what is expected under the null hypothesis's framework. Complete understanding of these hypotheses is essential to correctly interpreting the results of hypothesis tests.