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In the context of hypothesis testing, the null hypothesis serves as a starting point. It represents a statement of no effect or no difference, which we assume to be true until evidence suggests otherwise. Here, the null hypothesis, denoted as \(H_0\), posits that the average weight of the pineapple cans is 16 ounces. This can be formally written as \(H_0: \mu = 16\).

The null hypothesis is central to hypothesis testing because it provides a baseline for comparison. It is what we are testing against when we analyze our sample data. By examining the likelihood of our sample data under the assumption that the null hypothesis is true, we can make informed decisions about whether to reject or fail to reject this hypothesis.

The alternative hypothesis contrasts with the null hypothesis. It reflects the outcome that is expected or believed to be true. In this case, the alternative hypothesis, denoted as \(H_a\), suggests that the mean weight of the cans is greater than 16 ounces. This is expressed as \(H_a: \mu > 16\).

The role of the alternative hypothesis is pivotal as it guides the direction of the test. In this scenario, it implies a one-tailed test, focusing on whether the sample mean is significantly higher than the hypothesized population mean of 16 ounces. The evidence from our data will determine if we should support the alternative hypothesis over the null.

The significance level, symbolized by \(\alpha\), is a threshold used to decide when to reject the null hypothesis. It reflects our tolerance for risk in drawing incorrect conclusions. In most practical applications, a common significance level is 5%, or \(\alpha = 0.05\).

By setting \(\alpha = 0.05\), we specify that we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis (a Type I error). Thus, when the p-value, or probability of observing the test results under the null hypothesis, is less than \(0.05\), it indicates strong evidence against the null hypothesis, prompting us to reject it in favor of the alternative hypothesis.

The Z-test is a statistical procedure employed to determine if there is a significant difference between sample and population means. It is particularly suitable when the sample size is large, ensuring that the sample mean follows a normal distribution.

In our exercise, we use the Z-test to evaluate whether the mean of 50 pineapple cans is greater than the population mean of 16 ounces. The test statistic is calculated as \(Z = \frac{\bar{X} - \mu}{\frac{s}{\sqrt{n}}}\). Here, \(\bar{X}\) is the sample mean (16.05 ounces), \(\mu\) is the hypothesized mean (16 ounces), \(s\) is the sample standard deviation (0.03 ounces), and \(n\) is the sample size (50). Throughout this formula, the Z-score measures how far the sample mean deviates from the null hypothesis in units of standard error.

The p-value is a critical component in hypothesis testing, acting as a measure of the strength of evidence against the null hypothesis. It represents the probability of observing a test statistic as extreme as, or more extreme than, the one calculated assuming the null hypothesis is true.

In our scenario, the Z-score was calculated to be approximately 11.79. This high score indicates an extreme deviation from the expected value when assuming the null hypothesis holds true. The corresponding p-value is almost 0, which is far below the significance level of \(0.05\).

This nearly zero p-value implies that the sample mean of 16.05 ounces is statistically significant, leading us to conclude that it is implausible for the true mean to be 16 ounces. Therefore, we reject the null hypothesis, lending strong support to the alternative hypothesis that the mean weight of the cans is greater than 16 ounces.