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In the realm of statistics, a statistical hypothesis is a specific claim about a population parameter. In our example involving waitress tips, we are looking at whether introducing oneself increases tips by more than 50%. The parameters in question are the averages

• with introduction (\( \mu_1 \))
• without introduction (\( \mu_2 \))

A common way to test a hypothesis is to set up a null hypothesis (\( H_0 \)) and an alternative hypothesis (\( H_1 \)). The null hypothesis often represents a statement of "no effect" or "no difference," and in this situation, it suggests that the increase in tips does not reach 50%. In contrast, the alternative hypothesis proposes that the increase is indeed more significant. Framing these correctly is crucial as they guide the testing process.

The z-test is a statistical test used to determine whether there's a significant difference between sample means when the variances are known and the sample size is large (typically \( n > 30 \)). It's particularly useful when testing hypotheses about proportions, such as in our tipping exercise. The z-test helps determine how far the sample mean deviates from the null hypothesis mean, standardized by the variability in the data. For our problem, the z-test allows us to compare the difference in tip percentages with and without introductions against our expected outcome of a 50% increase. A high z-score would indicate a significant difference supporting our hypothesis, while a low z-score doesn't provide enough evidence.

A significance level is a threshold used to determine whether a hypothesis test result is statistically significant. It is denoted by \( \alpha \) and usually set at 0.05, meaning there's a 5% risk of concluding that a difference exists when there is none. In more straightforward terms, if our calculated test statistic exceeds the critical value associated with the 0.05 significance level, we reject the null hypothesis. This process implies a level of confidence that our findings are not due to random chance. However, if our calculated value doesn't surpass this critical threshold, we "fail to reject" the null hypothesis.

• Critical Value: Based on our exercise, the critical z-value is 1.645 for a one-tailed test.
• Decision Rule: Calculated value must be greater than 1.645 to be significant.

The test statistic is a crucial component of hypothesis testing, serving as the numerical summary of our data. It allows us to compare our sample data against the null hypothesis. In our tipping exercise, we computed the z-value to see if introducing oneself has the desired outcome. The formula used in the z-test for two independent samples is: \[ z = \frac{(\bar{x} - 1.5 \bar{y}) - 0}{\sqrt{\frac{s_1^2}{m} + \frac{(1.5s_2)^2}{n}}}\]Here:

• \( \bar{x} = 22.63 \) representing the average tip when the waitress introduces herself.
• \( \bar{y} = 14.15 \) representing the average tip without the introduction.
• \( s_1 \) and \( s_2 \) are the standard deviations.
• \( m \) and \( n \) represent the respective sample sizes.

The calculated z-value was approximately 0.827. As this is less than the critical z-value of 1.645, we fail to reject the null hypothesis, implying the introduction does not significantly increase tips by more than 50%.