9. Mickey D's In a study of the accuracy of fast food drive-through orders, McDonald's had 33 orders that were not accurate among 362 orders observed (based on data from \( Q S R \) magazine). Use a 0.05 significance level to test the claim that the rate of inaccurate orders is equal to \( 10 \% \). Does the accuracy rate appear to be acceptable?

To determine whether the rate of inaccurate McDonald's drive-through orders is acceptably low, we'll perform a hypothesis test for a population proportion. Here's a step-by-step analysis: ### **1. Define the Hypotheses** - **Null Hypothesis (\( H_0 \))**: The rate of inaccurate orders is equal to \( 10\% \). \[ H_0: p = 0.10 \] - **Alternative Hypothesis (\( H_a \))**: The rate of inaccurate orders is not equal to \( 10\% \). \[ H_a: p \neq 0.10 \] *This is a two-tailed test since we're checking for any significant difference from \( 10\% \).* ### **2. Collect and Summarize the Data** - **Sample Size (\( n \))**: 362 orders - **Number of Inaccurate Orders (\( X \))**: 33 - **Sample Proportion (\( \hat{p} \))**: \[ \hat{p} = \frac{X}{n} = \frac{33}{362} \approx 0.0912 \text{ or } 9.12\% \] ### **3. Calculate the Test Statistic** We'll use the **z-test for proportions**. \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Where: - \( p_0 = 0.10 \) (the hypothesized proportion) Plugging in the values: \[ z = \frac{0.0912 - 0.10}{\sqrt{\frac{0.10 \times 0.90}{362}}} = \frac{-0.0088}{\sqrt{\frac{0.09}{362}}} \approx \frac{-0.0088}{0.0158} \approx -0.559 \] ### **4. Determine the p-Value** Using the standard normal distribution: - **Find \( P(Z > |z|) \)** when \( z = -0.559 \). From standard normal tables or a calculator: \[ P(Z > 0.559) \approx 0.2877 \] Since it's a two-tailed test: \[ \text{p-value} = 2 \times 0.2877 \approx 0.5754 \] ### **5. Make a Decision** - **Significance Level (\( \alpha \))**: 0.05 - **Compare p-value with \( \alpha \)**: \[ 0.5754 > 0.05 \] - **Decision**: **Fail to reject** the null hypothesis. ### **6. Conclusion** There is **insufficient evidence** at the \( 0.05 \) significance level to conclude that the rate of inaccurate McDonald's drive-through orders differs from \( 10\% \). **Interpretation for Acceptability**: Since the observed inaccurate rate is approximately \( 9.12\% \), which is not significantly different from the claimed \( 10\% \), the accuracy rate **appears to be acceptable** based on this study. ### **Summary** - **Test Used**: Two-tailed z-test for population proportion. - **Test Statistic**: \( z \approx -0.559 \) - **p-Value**: \( \approx 0.5754 \) - **Conclusion**: Fail to reject \( H_0 \); the accuracy rate is acceptable.