The margarita is one of the most common tequila-based cocktails, made with tequila mixed with Triple Sec and lime or lemon juice, often served with salt on the glass rim. A common ratio for a margarita is 2:1:1, which includes 50% tequila, 25% Triple Sec, and 25% fresh lime or lemon juice. A manager at a local bar is concerned that the bartender uses incorrect proportions in more than 50% of margaritas. The manager secretly observes the bartender and finds that the bartender used the correct proportions in only 14 out of 40 margaritas. Can we conclude that the manager’s suspicion is justified? Use α = 0.05. Let *p* represent the population proportion of incorrectly made margaritas.
**c.**  Calculate the test statistic.

Test statistic

**d.**  Calculate the *p*-value.

*p*-value

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.

The margarita is one of the most common tequila-based cocktails, made with tequila mixed with Triple Sec and lime or lemon juice, often served with salt on the glass rim. A common ratio for a margarita is 2:1:1, which includes 50% tequila, 25% Triple Sec, and 25% fresh lime or lemon juice. A manager at a local bar is concerned that the bartender uses incorrect proportions in more than 50% of margaritas. The manager secretly observes the bartender and finds that the bartender used the correct proportions in only 14 out of 40 margaritas. Can we conclude that the manager’s suspicion is justified? Use α = 0.05. Let p represent the population proportion of incorrectly made margaritas. Calculate the test statistic and p-value.

Learn how to perform a hypothesis test to evaluate whether a bartender uses incorrect proportions in margaritas over 50% of the time. This solution covers hypothesis formulation, test statistic calculation, and p-value interpretation, using a sample of 40 margaritas with α = 0.05.

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