Confidence Interval For The Population Mean

Confidence Interval For The Population Meanproject Descrip

The present study shows data for delivery times of international shipments for a logistics company. The company promotes its international delivery as a service that takes up to 7 days. After receiving multiple complaints that the delivery time is exceeding 7 days, the quality assurance team decided to collect data from a random sample to analyze the delivery times and decide if they should adjust the delivery time the company promises customers with. We will find the average and standard deviation for the delivery times for the random sample of shipments. We will find values from the probability table using excel functions to construct confidence intervals using different confidence levels.

We will use the confidence intervals to make a conclusion about the population. We will also analyze the width of the confidence levels. Assume that the distribution of the delivery times is normal and the sample is randomly selected.

Paper For Above instruction

The assessment of delivery times for international shipments is crucial for logistics companies aiming to uphold customer satisfaction and maintain competitive service standards. In this context, the use of confidence intervals provides a statistical foundation for evaluating whether the company's promise of delivery within 7 days holds true for the broader population of shipments, based on sample data. This study demonstrates how to compute and interpret confidence intervals at different confidence levels, highlighting their implications for quality assurance and decision-making.

Initially, data collected from a randomly selected sample of shipment delivery times serves as the basis for analysis. The sample mean, standard deviation, and size are fundamental descriptive statistics computed using Excel functions. These metrics not only summarize the sample data but also facilitate the construction of confidence intervals that estimate the population mean delivery time with a specified degree of certainty.

Constructing a confidence interval involves selecting the appropriate probability distribution table. Since the sample size is typically small, and the population standard deviation is unknown, the Student's t-distribution is suitable. Using Excel’s T.INV.2T function, critical t-scores for 90% and 99% confidence levels are determined. The margin of error, derived via Excel’s CONFIDENCE.T function, quantifies the maximum expected deviation of the sample mean from the true population mean at each confidence level.

The upper and lower bounds of each confidence interval are calculated by adding and subtracting the margin of error from the sample mean. These bounds provide a range within which the true average delivery time for all shipments is likely to fall with the specified confidence. A key part of the analysis involves interpreting these intervals: if the interval contains the promised delivery time of 7 days, the company cannot confidently assert that their promise is unreasonable based on the sample data. Conversely, if the interval exceeds 7 days, it suggests that the company's delivery promise may need reassessment.

Comparing the widths of the confidence intervals at 90% and 99% confidence levels reveals that higher confidence levels produce wider intervals. This increase in width reflects greater certainty but also indicates less precision regarding the estimate of the population mean. Such insights assist the quality assurance team in understanding the trade-offs between confidence level and estimate precision.

The decision to modify the company's promise depends on whether the confidence intervals, particularly at higher confidence levels, remain above the 7-day threshold. If the intervals consistently suggest that the average delivery time exceeds 7 days, the team should consider recommending a revision of the delivery promise to manage customer expectations and improve service commitments. Conversely, if the intervals mostly fall within or include 7 days, no immediate change may be necessary.

To enhance the accuracy of the estimation, the team could consider increasing the sample size, which would narrow the confidence intervals by reducing the margin of error. Alternatively, refining data collection methods or reducing variability in delivery times would also contribute to more precise estimates, supporting more definitive business decisions.

In conclusion, the use of confidence intervals in analyzing shipment delivery times provides a statistical basis for operational and strategic decisions. They help quantify the uncertainty associated with sample estimates and guide management on whether to uphold, adjust, or review their delivery commitments based on empirical data.

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