Chicken Delight Claims That 81 Percent Of Its Orders Are Del

Chicken Delight Claims That 81 Percent Of Its Orders Are Delivered Wit

Chicken Delight asserts that 81 percent of its orders are delivered within 10 minutes of placement. To evaluate this claim, a sample of 80 orders was analyzed, and it was found that 61 orders were delivered within the promised time. Using this data, we are tasked with testing, at a significance level of 0.10, whether the true proportion of orders delivered within 10 minutes is less than 81 percent.

The core question involves conducting a hypothesis test for a population proportion. Specifically, the null hypothesis (H₀) posits that the true proportion of timely deliveries is 0.81, consistent with the company's claim. The alternative hypothesis (H₁) suggests that this proportion is less than 0.81, indicating that fewer orders are delivered within the promised time than claimed.

Paper For Above instruction

Introduction

The punctuality of order delivery is a critical aspect of customer satisfaction in the food delivery industry. Accurate measurement and testing of delivery times against company claims are essential for maintaining credibility and improving service quality. In this context, Chicken Delight claims that 81% of its orders are delivered within 10 minutes. To verify this claim, a statistical hypothesis test is conducted based on a sample of 80 orders, of which 61 were delivered on time. This paper discusses the process and results of this hypothesis test, aiming to assess whether the actual delivery rate is significantly lower than the claimed rate at a 10% significance level.

Methodology

The hypothesis test employed is a one-proportion z-test, appropriate for analyzing whether a sample proportion differs significantly from a claimed population proportion. The sample proportion (p̂) was calculated as:

p̂ = Number of successful deliveries / Total deliveries = 61 / 80 = 0.7625.

The null hypothesis (H₀) assumes the true proportion (p) is 0.81:

H₀: p = 0.81

The alternative hypothesis (H₁) posits that the true proportion is less than 0.81:

H₁: p

This directional hypothesis corresponds to testing whether delivery performance is below the company’s claimed level.

The significance level (α) is set at 0.10, which indicates a willingness to accept a 10% probability of Type I error—that is, falsely rejecting the null hypothesis when it is true.

The test statistic for the proportion z-test is calculated by:

z = (p̂ - p₀) / √(p₀(1 - p₀) / n)

where p₀ is the claimed proportion (0.81), and n is the sample size (80).

Calculating the z-score:

z = (0.7625 - 0.81) / √(0.81 × 0.19 / 80)

= -0.0475 / √(0.1539 / 80)

= -0.0475 / √0.001924

= -0.0475 / 0.04386

≈ -1.082

Next, the critical value corresponding to a left-tailed test at α = 0.10 is approximately -1.28.

Results and Interpretation

Since the calculated z-value of -1.082 is greater than the critical z-value of -1.28, we fail to reject the null hypothesis at the 10% significance level. This indicates that there is not enough evidence to conclude that the true proportion of timely deliveries is less than 81%. In other words, based on this sample, the data are consistent with the company's claim, although the sample proportion (0.7625) is slightly lower than 0.81.

It is important to note that failing to reject the null hypothesis does not prove the null is true; it simply suggests that, at the 10% significance level, the evidence is insufficient to conclude a decline in delivery performance.

Discussion on Limitations and Implications

While the sample size provides useful insights, it may not be large enough to detect small but meaningful differences in delivery rates with high confidence. Additionally, real-world factors such as traffic conditions, peak hours, and order volume could influence delivery times, implying that the sample may not fully capture the variability in delivery performance.

From a managerial perspective, the results suggest that Chicken Delight's delivery service approximately aligns with its claimed performance, though the observed proportion is somewhat lower. The company might consider ongoing monitoring and larger samples to more precisely estimate delivery performance over time.

Conclusion

In summary, the statistical analysis indicates that there is no significant evidence at the 10% significance level to reject Chicken Delight’s claim that 81% of orders are delivered within 10 minutes. Although the sample proportion was slightly lower, the difference was not statistically significant enough to affirm that fewer than 81% of deliveries are timely. Continuous assessment and larger sample analyses are recommended for more definitive conclusions.

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