Analyzing Daily Cookie And Smoothie Sales: Symmetry, Outlier ✓ Solved

Analyzing Daily Cookie and Smoothie Sales Symmetry Outliers and Probabilities

Analyzing Daily Cookie and Smoothie Sales: Symmetry, Outliers, and Probabilities

The provided data reflects the daily sales figures for cookies and smoothies outside over a specific period. The aim is to interpret the distribution characteristics of these sales, such as skewness and symmetry, and utilize statistical measures to estimate probabilities and identify outliers. This analysis offers valuable insights for forecasting, inventory management, and strategic planning in the context of food service sales.

Firstly, understanding whether the daily sales of cookies and smoothies are symmetric or skewed helps determine the appropriate statistical models and forecasts. The given skewness value, close to zero (0.044), indicates an approximately symmetric distribution for the sales data. Symmetry in the data suggests that the mean, median, and mode are approximately equal, underpinning the use of parametric methods predicated on normal distribution assumptions.

Understanding Distribution Characteristics and Central Measures

The dataset provides key descriptive statistics, including the mean (218.47), median (206), and mode (166). The mean and median are close, and the skewness statistic indicates minimal skewness, confirming the data's approximate symmetry. The mode, being lower than the mean, suggests the most frequently occurring daily sales tend to be below the average, which often occurs in symmetric distributions where the data are balanced but may have slight variations.

Estimating the Range Covering 95% of Sales Days

According to the empirical rule (or 68-95-99.7 rule), for a symmetric, approximately normal distribution, about 95% of data will fall within two standard deviations of the mean. Given the standard deviation of approximately 74, the interval can be calculated as:

  • Lower limit = Mean - 2 × Standard Deviation = 218.47 - 2 × 74 ≈ 218.47 - 148 ≈ 70.47
  • Upper limit = Mean + 2 × Standard Deviation = 218.47 + 148 ≈ 366.47

Roundly, we can expect that on about 95% of days, cookie sales will be between approximately 70 and 366 cookies, aligning with the 'between ___ and ___' statement provided.

Identifying Outliers in Smoothie Demand

Outliers are demands significantly lower or higher than typical values. Using the standard deviation and the mean, demands below two standard deviations from the mean (here, less than about 70) or above the upper bound (about 366) could be considered outliers.

Specifically, demand less than 70 or greater than 366 will be classified as outliers, which can be critical for operational planning and inventory control to handle exceptional days.

Calculating the Probability that Daily Smoothie Demand Exceeds a Threshold

The problem states there is a 10% chance that daily smoothy demand will exceed a certain value, which has been identified as approximately 317 smoothies. This aligns with the properties of the normal distribution where the top 10% of demand values exceed the 90th percentile.

Using the z-score formula:

z = (X - mean) / standard deviation

Plugging in the values:

z = (317 - 218.47) / 74 ≈ 98.53 / 74 ≈ 1.33

The cumulative probability for z ≈ 1.33 corresponds approximately to 0.9082, implying about 90.82% of demands are below 317. Conversely, roughly 10% of days will have demand exceeding this level, confirming the statement in the provided data.

Estimating the Probability of Selling at Least 600 Cookies

The dataset indicates the 70th percentile for cookie sales is approximately 600. This percentile point implies that 70% of days have sales below 600 cookies, and the remaining 30% exceed it.

To find the probability of selling at least 600 cookies, we look at the complement of the 70th percentile:

P(sales ≥ 600) ≈ 30%

Assuming the distribution is normal, the z-score for a sale of 600 cookies:

z = (600 - 218.47) / 74 ≈ 381.53 / 74 ≈ 5.15

Such a high z-score corresponds to a probability practically close to zero, demonstrating the rarity of such high sales figures and confirming the percentile-based probability estimate. Thus, there's roughly a 30% chance of daily sales exceeding 600 cookies.

Implications for Business Operations

Understanding these statistical characteristics informs inventory and staffing decisions. Knowing that 95% of smoothy demands fall within 70 to 366 allows for stock planning within this range, reducing waste and stockouts. Identifying outliers ensures preparedness for exceptional demand days. Recognizing the probabilities associated with extreme sales assists in strategic planning to meet customer demand without overstocking or underpreparing.

Conclusions

Overall, the sales data for cookies and smoothies reveal approximate symmetry, justified by the near-zero skewness and the close alignment of mean, median, and mode. Using the empirical rule facilitates effective estimation of demand intervals, identification of outliers, and probability calculations for demand exceeding specific thresholds. Such insights are vital for optimizing supply chain efficiency in food service operations, ensuring adequate resource allocation, and maintaining customer satisfaction.

References

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