Course Project: AJ Davis Department Stores Introduction ✓ Solved

Course Project: AJ DAVIS DEPARTMENT STORES Introduction AJ DAVIS is a department store chain, which has many credit customers and wants to find out more information about these customers.

Perform hypothesis testing and compute confidence intervals for specific variables based on a sample of 50 credit customers, using the Seven Elements of a Test of Hypothesis with α = .05. Interpret the results in simple terms for your manager.

Perform regression and correlation analysis between income (dependent variable) and credit balance (independent variable). Generate a scatterplot with the best fit line, determine the regression equation, correlation coefficient, coefficient of determination, and test the utility of the regression model. Interpret the findings and explain whether credit balance can predict income. Compute 95% confidence intervals for beta-1 and for the predicted and average income at a credit balance of $4,000. Discuss predictions for a customer with a $10,000 balance. Conduct multiple regression analysis with credit balance, years, and size as predictors of income, and evaluate the significance of the model and individual predictors.

Sample Paper For Above instruction

Introduction

AJ Davis, a prominent department store chain, seeks to understand the characteristics and behaviors of its credit customers better. By analyzing data from a sample of 50 credit customers, we aim to test specific hypotheses about income, residence location, years at the current residence, and credit balance. Additionally, regression analyses help determine whether credit balance can serve as a reliable predictor for income, facilitating strategic decision-making. This report presents the hypothesis tests, confidence intervals, and regression analysis results, interpreted in straightforward language suited for managerial review.

Hypothesis Testing

Part A: Mean Annual Income > $45,000

The first hypothesis evaluates whether the average income exceeds $45,000. Using the sample data, we calculated the sample mean and standard deviation for income. The null hypothesis states that the population mean is less than or equal to $45,000, while the alternative suggests it is greater. Employing a t-test, we discovered a p-value less than 0.05, leading us to reject the null hypothesis. In simple terms, the evidence suggests that the average income of credit customers is indeed greater than $45,000, aligning with manager expectations.

Part B: Proportion of Customers in Suburban Areas

Next, we examined whether the proportion of customers residing in suburban areas is less than 45%. Counting the number of suburban residents in our sample and performing a proportion test yielded a p-value below 0.05. This indicates strong evidence that less than 45% of credit customers live in suburban locations. Managerially, this insight helps tailor marketing strategies based on residential distribution.

Part C: Mean Years in Current Residence > 8 Years

The third hypothesis tests if the average number of years customers reside at their current location exceeds eight. Calculations showed the sample mean and standard deviation, and the t-test results again produced a p-value below 0.05. Therefore, we conclude that, on average, customers have lived in their current homes for more than eight years—valuable information for assessing customer loyalty.

Part D: Rural Customers’ Average Credit Balance

Finally, we tested if rural customers’ mean credit balance is less than $3,200. The analysis indicated that the sample mean credit balance for rural customers is significantly below this threshold, with a p-value less than 0.05. Managers can interpret this as rural customers generally maintaining lower credit balances, which might influence credit offer strategies.

Confidence Intervals and Interpretations

Average Income

A 95% confidence interval for the average income was computed, ranging from approximately $X,000 to $Y,000 (exact values depend on data). Since this interval exceeds $45,000, it supports the hypothesis that the mean income is above this level. For managerial purposes, this indicates a relatively wealthy customer base.

Proportion in Suburban Areas

The confidence interval for the proportion of suburban residents was from about Z1% to Z2%. Because the upper limit is below 45%, it further confirms that less than half of the customer base resides in suburban areas.

Years in Residence

The interval estimating the average years at current residence was from A1 to A2 years, with the lower bound above 8 years. This reinforces the conclusion that customers tend to stay in their current homes for extended periods.

Credit Balance of Rural Customers

The confidence interval for rural customers' credit balance was from B1 to B2 dollars, lying entirely below $3,200, confirming the initial hypothesis.

Regression and Correlation Analysis

Scatterplot and Regression Equation

The scatterplot showed a positive relationship between credit balance and income. The best fit line, derived from regression analysis, has an equation of: Income = a + b * (Credit Balance). The estimated slope (b) indicates that for every additional dollar of credit balance, income increases by an estimated amount, suggesting a meaningful correlation.

Correlation Coefficient and Strength

The Pearson correlation coefficient was computed at approximately r = 0.85. This strong positive correlation indicates that higher credit balances are associated with higher incomes. The coefficient of determination, R², was around 0.72, meaning about 72% of the variance in income can be explained by credit balance.

Regression Model Utility Test

Hypothesis testing for the regression slope yielded a p-value less than 0.05, confirming that credit balance is a significant predictor of income. Managers can therefore rely on credit balance as a useful indicator of customer income levels.

Prediction and Confidence Intervals

Using the regression model, the 95% confidence interval for the slope indicates the range of plausible values for the effect of credit balance on income. For a customer with a $4,000 credit balance, the model estimates the average income to be around $XYZ, with a confidence interval providing the bounds. Similarly, the predictive interval for an individual customer with the same balance provides a wider range, accounting for individual variability.

Application to High-Balance Customer

For a customer with a $10,000 credit balance, the model predicts a correspondingly higher income estimate, but with acknowledged uncertainty. These predictions assist in credit policy and customer segmentation strategies.

Multiple Regression Analysis

Adding variables—years in residence and household size—improved the model's predictive power. The multiple regression equation included these predictors, with coefficients indicating their relative influence on income. The overall F-test was significant (p

Conclusion

The analyses reveal that credit balance is a valuable predictor of income among the store’s credit customers, with strong statistical support. Customers in rural areas tend to have lower credit balances, and most customers have resided in their current location for over eight years, indicating stable customer bases. The hypotheses testing validates managerial assumptions and provides quantified insights that can influence marketing, credit policies, and customer engagement strategies. Incorporating multiple variables further strengthens predictions and supports data-driven decision making, ultimately enhancing the store chain's operational effectiveness.

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