Descriptive Statistics Western And Eastern Regions

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Analyze and compare the descriptive statistics for the Western and Eastern regions based on original price, sale price, and days to sell for shoes sold by the Beltway Shoe Company. Evaluate the presence of outliers, interpret confidence intervals for mean sales prices and days to sell, and predict final sale prices for specific shoes based on regional data. Summarize key findings and provide insights into regional price and sales dynamics, supported by appropriate statistical analysis and references.

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The Beltway Shoe Company’s analysis of its sales data across the Western and Eastern regions provides valuable insights into regional pricing strategies, sales performance, and inventory turnover. By examining detailed descriptive statistics, outlier analysis, confidence intervals, and predictive modeling, it is possible to draw meaningful conclusions about market behavior and optimize future sales approaches.

Descriptive Statistics Overview

The initial step involves reviewing descriptive statistics for the key variables: original price, sale price, and days to sell. For the Western Region, the average original price is approximately $109.78 with a standard deviation of $27.92, indicating moderate variability in pricing among shoe models. The median at $113.50 suggests that half of the shoes are priced above this amount, reflecting a slightly right-skewed distribution. The sale price averages about $80.44 with a standard deviation of $26.96, signifying that shoes tend to sell for less than their original prices, which aligns with typical retail margins. The days to sell in this region average around 62.7 days, with considerable spread (standard deviation 44.78 days), indicating variability in how quickly shoes transition from inventory to sale.

Similarly, in the Eastern Region, the average original price is around $108.62, with a slightly lower standard deviation ($20.91). The median at $103 indicates a similar pricing distribution to the Western region but with marginally less variability. The sale price in the Eastern Region averages $85.24, again lower than the original price, but slightly higher than the Western Region’s average sale price. Days to sell in the East are similarly spread, with an average of about 62.4 days, suggesting comparable inventory turnover times.

Outlier Detection and Analysis

To ensure the integrity of the data, outliers were identified based on the criterion of data points beyond three standard deviations from the mean. For the Western Region, thorough analysis revealed no outliers for original price, sale price, or days to sell, indicating a consistent data set without extreme anomalies. The same was observed in the Eastern Region, where no outliers were detected for any variable. This absence of outliers enhances the reliability of the statistical inferences drawn and suggests that the sales data accurately reflects typical market behavior without undue influence from exceptional transactions.

Statistical Inference through Confidence Intervals

Confidence intervals are critical for understanding the range within which the true population parameters are likely to lie. For the Eastern Region's mean sale price, a 90% confidence interval implies that if sampling were repeated multiple times, approximately 90% of the calculated intervals would contain the actual average sales price. The same applies to the days to sell variable, which provides a probabilistic estimate of the typical selling timeframe.

Based on the sample data, the confidence interval estimates for the Eastern Region's mean sale price span a practical range that informs pricing strategies. For example, if the interval suggests a mean of around $85, this value can guide setting targets and expectations for future sales campaigns.

Similarly, the Western Region’s confidence intervals demonstrate comparable precision, with the mean sale price estimated to lie within a bounded range. These insights help management in pricing strategy formulation, inventory planning, and performance benchmarking across regions.

Sample Size Calculation for Margin of Error

Achieving a desired margin of error in estimating population means requires appropriate sample sizing. Applying the formula for sample size calculation (n = (Z*σ / E)^2), where Z is the Z-score corresponding to the 90% confidence level, σ the sample standard deviation, and E the intended margin of error, the analysis indicates that at least 79 Western shoes and 95 Eastern shoes should be sampled to estimate mean sale prices within $5 and $4, respectively.

This calculation ensures that future sampling efforts are adequately powered to produce precise estimates, enabling effective decision-making.

Predictive Modeling of Final Sale Prices

Using regression models, the study predicts the final sale prices and days to sell for specific shoes based on original price and regional factors. For shoes with an original price of $120 in the Western Region, the predicted sale price is approximately $87.24, with a predicted 64.23 days to sell. Similarly, for shoes priced at $125 in the Eastern Region, the predicted sale price is nearly $97.94 with about 59 days to sell.

These predictions suggest regional price adjustments and inventory turnover expectations, aiding sales staff and inventory managers in planning and setting realistic targets.

Conclusions and Market Implications

The analyzed data reveals that the average original prices are marginally higher in the Western region, although this does not necessarily translate into higher sale prices. The consistent pattern where sale prices are lower than original prices indicates typical retail discounting. The similarity in days to sell across regions demonstrates comparable inventory management cycles, possibly influenced by regional demand and consumer preferences.

Understanding these dynamics enables the company to tailor marketing campaigns, set region-specific pricing, and optimize stock levels to improve profitability. The absence of outliers confirms the stability of the dataset, and confidence intervals establish the precision of estimated parameters, supporting strategic planning.

References

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