Quantitative Methods - I June 2026
Q.1: After surveying a sample of 100 new students, the university finds that 40 indicate a preference for Chinese food. The student affairs office wants to determine if this marks a meaningful shift from prior years' 30% rate, guiding future dining options. They require a clear, defensible statistical decision process rather than relying on intuition or anecdote. How should the university use z-scores and standard error calculations to identify whether the proportion of students preferring Chinese food in the new batch is significantly different from the historic 30%? Outline the steps and justify the statistical choices involved.
Answer:
Introduction:
Using only your gut feelings to make decisions that could impact your organization will often lead you to make poor judgment calls. For instance, the university in question wants to find out if the increased consumer preference toward Chinese food from incoming freshmen is a true increase in consumer preference or just a fluke based on how random sampling works. To answer this type of question statistically, the university would use a statistical hypothesis test as a structured and objective process to work through questions like the one just quoted here. The way that the university could compare the difference between their historical benchmark (30% true proportion) and the result of their survey conducted with 100 freshmen, (40% preference for Chinese food). At first glance, one would think that the difference (10%) is significant enough to be considered a trend; however, random sampling can produce results with similar differences regularly, so the university must conduct their own statistical test to determine if this slight increase in preference (between 30% and 40%) can be viewed as a statistically significant difference or coincidence. Since the university's sample size is large enough, the university should utilize z-testing when calculating whether an increase of 10% between previous and current preferences for China food is statistically significant.
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Q.2 (A): A large retail chain uses a contingency table to analyze the shopping habits of its customers based on gender and number of purchases per week. Despite initial insights from joint and marginal probabilities, the management is debating how much attention should be paid to conditional probabilities for segmenting targeted marketing campaigns. There is also internal disagreement whether the events (gender and number of purchases) are independent or not, especially when tailoring cross-selling strategies. This decision impacts both budget allocation and the accuracy of campaign targeting. Critically evaluate the advantages and limitations of relying on marginal, joint, and conditional probabilities for customer segmentation in this scenario. Assess whether assuming independence or dependence between gender and purchasing behavior improves decision-making, and justify which approach the retail chain should adopt for optimal campaign effectiveness.
Answer:
Introduction:
For retail chains to develop an appropriate marketing approach that works well with different types of customers, they need to identify your customers into distinct segments through customer segmentation. To help summarize the relationship between two separate variables (i.e., gender and the number of purchases made weekly), a contingency table will provide the information needed to calculate the probability of each variable occurring. The decision made by management regarding whether to use marginal, joint, or conditional probabilities and whether the two variables are independent or dependent will impact how well the target customer is being reached and how much of the marketing budget is utilized.
Q.2 (B): A national retail chain operates 250 stores across different regions. The average monthly sales per store follow a normal distribution with a mean of $150,000 and a standard deviation of $20,000. Management wants to estimate the probability that a randomly selected store generates monthly sales exceeding $180,000. The results will be used to assess how realistic their premium store classification target is.
Using the normal distribution framework:
1. Calculate the probability that a store earns more than $180,000 in a month.
2. Interpret the result in a managerial context.
3. Based on your findings, comment on whether the premium classification threshold appears too strict or reasonable.
Answer:
Introduction:
Understanding how much sales fluctuate is critical for achieving accurate sales performance goals in stores, as the basis for developing realistic goals on future sales projections. Because the sales of stores are normally distributed, statistical formulas such as the z-score can provide the estimated probability of hitting a certain level of monthly sales. Here, management is trying to evaluate how probable it would be for a store to have monthly sales over $180,000 in comparison to the mean of $150,000. This will help to evaluate if their classification of store as “premium” is reasonable or unrealistic.