How To Decrease Delivery Time For Packages

To Decrease The Amount Of Time It Takes To Deliver Packages A Deli

To decrease the amount of time it takes to deliver packages, a delivery company purchased computer software that finds the optimal route for making deliveries based upon the input of the package destinations. In the past the average amount of time it took to complete deliveries was 9.4 hours with a standard deviation of 0.8 hours. Since purchasing the software the average delivery time over twenty delivery days was 8.6 hours. At the 0.05 level of significance, test whether the software package has reduced the average time it takes to complete deliveries.

Over the past few years, banks have strongly promoted online banking and the paying of bills online. In 2003, about 50% of internet users paid bills online. In a recent survey of 250 internet users, 150 said that they pay bills online. At the 5% level of significance, test whether the proportion of internet users paying bills online is now more than 50%.

Paper For Above instruction

Part 1: Testing the Effectiveness of a Delivery Software on Delivery Time

The company’s initiative to adopt new routing software aims to enhance operational efficiency by reducing delivery times. To evaluate the impact of this technology, hypothesis testing provides a systematic approach. Initially, the null hypothesis (H₀) posits that there is no reduction in delivery time due to the software, i.e., the mean delivery time remains 9.4 hours. Conversely, the alternative hypothesis (H₁) suggests that the software has indeed decreased the delivery time, indicating that the mean is less than 9.4 hours.

Given data: The population mean delivery time (μ₀) is 9.4 hours, standard deviation (σ) is 0.8 hours, sample mean (x̄) from 20 days is 8.6 hours. The sample size (n) is 20. Since the population standard deviation is known, a z-test is appropriate. The test statistic is calculated as:

z = (x̄ - μ₀) / (σ / √n) = (8.6 - 9.4) / (0.8 / √20) ≈ -4.4721.

At a significance level of α = 0.05, the critical value for a one-tailed test is approximately -1.645. Our computed z value (-4.4721) is less than -1.645, indicating strong evidence to reject the null hypothesis. Consequently, it is statistically significant to conclude that the software has reduced the mean delivery time.

Part 2: Testing Changes in Internet Users Paying Bills Online

Historical data indicates that in 2003, roughly 50% of internet users paid their bills online. A recent survey of 250 internet users found that 150 pay bills online. To assess whether this proportion has increased beyond 50%, we perform a hypothesis test about a population proportion.

Null hypothesis (H₀): p = 0.50

Alternative hypothesis (H₁): p > 0.50

Sample proportion (p̂) = 150 / 250 = 0.6

The standard error (SE) is computed as:

SE = √[p₀(1 - p₀) / n] = √[0.5 * 0.5 / 250] ≈ 0.0316.

The test statistic (z) is calculated as:

z = (p̂ - p₀) / SE = (0.6 - 0.5) / 0.0316 ≈ 3.1623.

At α=0.05, the critical z value for a one-tailed test is approximately 1.645. Since 3.1623 > 1.645, we reject H₀ and conclude that there is significant evidence to suggest that more than 50% of internet users pay bills online now.

This indicates a positive trend in online bill payments among internet users, aligning with the promotional efforts of banks over recent years.

Conclusion

Through hypothesis testing, the company's software demonstrates a significant reduction in delivery times, substantiating its effectiveness. Similarly, the increase in the proportion of internet users paying bills online is statistically significant, confirming a behavioral shift possibly driven by the banks' promotional campaigns. These analyses underscore the importance of statistical inference in evaluating technological and societal changes, facilitating informed decision-making for organizations and policymakers.

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