To determine if the manager's suspicion is justified, we can perform a hypothesis test on the population proportion of incorrectly made margaritas. Let's define our hypotheses:

• Null hypothesis $(H_0)$: $p = 0.5$
• Alternative hypothesis $(H_a)$: $p > 0.5$

Here, $p$ represents the true proportion of incorrectly made margaritas, and we are conducting a one-tailed test with a significance level of $\alpha = 0.05$.

Step 1: Define sample statistics and calculate sample proportion

The bartender used the correct proportions in 14 out of 40 margaritas, so the number of incorrectly made margaritas is: $x = 40 - 14 = 26$ Thus, the sample proportion of incorrectly made margaritas is: $\hat{p} = \frac{26}{40} = 0.65$

Step 2: Calculate the test statistic $z$

The test statistic for a population proportion is calculated as: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$ where:

• $\hat{p} = 0.65$
• $p_0 = 0.5$ (the hypothesized population proportion)
• $n = 40$ (sample size)

Plugging in these values: $z = \frac{0.65 - 0.5}{\sqrt{\frac{0.5 \times (1 - 0.5)}{40}}} = \frac{0.15}{\sqrt{\frac{0.25}{40}}} = \frac{0.15}{\sqrt{0.00625}} = \frac{0.15}{0.07906} \approx 1.897$

Step 3: Calculate the p-value

For a one-tailed test, the p-value is the probability that $Z$ is greater than or equal to the calculated test statistic under the standard normal distribution.

Using $z = 1.897$, the p-value can be looked up in a standard normal table or calculated, which gives: $p\text{-value} \approx 0.029$

Conclusion

Since the p-value $\approx 0.029$ is less than the significance level $\alpha = 0.05$, we reject the null hypothesis. This suggests there is enough evidence to support the manager's suspicion that the bartender uses incorrect proportions in more than 50% of margaritas.

Would you like further clarification or a step-by-step breakdown of any part of this calculation?

---

1. What does rejecting the null hypothesis mean in practical terms for the bar manager?
2. How is the p-value interpreted in hypothesis testing?
3. How would the conclusion change if the significance level were set to $\alpha = 0.01$?
4. How might sample size impact the reliability of this hypothesis test?
5. Could a two-tailed test be applied in this scenario, and if so, how would it affect the interpretation?

Tip: When conducting hypothesis tests, always define hypotheses and identify the correct test type (one-tailed or two-tailed) based on the research question.