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In hypothesis testing, the null hypothesis (\( H_0 \)) is a statement that there is no effect or difference between groups or variables. It suggests a default position that any kind of difference or effect you see in a set of data is not significant and could be due to chance. In our case, the null hypothesis claims that the average waiting time at MacBurger is exactly 3 minutes. The null hypothesis is critical because it provides a benchmark against which we can measure our evidence. If our data are inconsistent with the null hypothesis, we can reject it in favor of the alternative hypothesis. This forms the foundational step in hypothesis testing and directs subsequent analysis. Remember, rejecting the null hypothesis does not prove the alternative hypothesis is true, but rather that it is a better explanation based on the evidence.

The significance level, commonly denoted as \( \alpha \), is the probability of rejecting the null hypothesis when it is actually true. It represents a threshold of risk that researchers are willing to take. In our scenario, the significance level is set at 0.05, meaning there is a 5% chance of wrongly concluding that the average waiting time is less than 3 minutes when it is not.A lower significance level means stricter criteria for rejecting the null hypothesis, while a higher significance level is more lenient. Choosing an appropriate significance level is essential in balancing the risk of making Type I errors (false positives) and Type II errors (false negatives). Usually, 0.05 is a conventional choice in many fields, indicating a moderate balance between caution and discovery.

The Z-test is a statistical tool used to determine whether there is a significant difference between the means of a sample and a population. It is applicable when the sample size is large (\( n > 30 \)), and the data are approximately normally distributed.For the MacBurger case, we apply the Z-test to assess whether the average waiting time of 2.75 minutes in a sample of 50 customers truly reflects a general reduction from the claimed 3 minutes. The Z-test formula is:\[ Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]Where:

• \( \bar{x} \) is the sample mean (2.75 minutes).
• \( \mu \) is the population mean (3 minutes).
• \( \sigma \) is the standard deviation (1 minute).
• \( n \) is the sample size (50).

By calculating the Z-value, we determine how many standard deviations away the sample mean is from the population mean. A significant Z-value indicates evidence against the null hypothesis.

The critical value is a threshold that determines the boundary between the acceptance region and the rejection region of the null hypothesis. It is derived based on the significance level and the nature of the test (one-tailed or two-tailed). For a one-tailed test with \( \alpha = 0.05 \), the critical value is \(-1.645\).In our test of the MacBurger waiting time, we compare the calculated Z-value with this critical value. If the Z-value is less than the critical value (in the case of a left-tailed test), then we have sufficient evidence to reject the null hypothesis. Here, since our Z-value is \(-1.77\), which is less than \(-1.645\), we confidently reject the null hypothesis, indicating that the mean waiting time is statistically less than 3 minutes at this significance level.Understanding critical values helps in making informed decisions based on statistical evidence and the predefined risk level.