The Scatterplot Has A Positive Correlation As X Increases

The Scatterplot Has A Positive Correlation As X Increases So Does Y

The assignment involves analyzing the relationship between shelf space allocated to pet food and weekly pet food sales based on data collected from a sample of stores. The primary goal is to determine whether a linear relationship exists between these two variables, using statistical techniques such as correlation, regression analysis, hypothesis testing, and graphical evaluation of assumptions. The analysis includes developing a regression model, testing significance of coefficients, verifying model assumptions, and interpreting confidence and prediction intervals.

Paper For Above instruction

The relationship between shelf space allocated to pet food products and the corresponding weekly sales is a critical factor influencing marketing strategies and inventory management in retail settings. Understanding whether an increase in shelf space correlates with higher sales enables managers to optimize store layouts for maximum profitability. This study employs quantitative statistical analysis to determine and interpret the potential linear relationship between these variables using data collected from fifteen stores.

Initially, the problem is defined as evaluating whether the amount of shelf space (measured in square footage or linear feet) significantly impacts weekly pet food sales (measured in hundreds of dollars). The hypothesis testing begins with the null hypothesis (H0) that there is no linear relationship between shelf space and sales (p=0), against the alternative hypothesis (H1) that a positive correlation exists (p ≠ 0). The significance level (α) is set at 0.01 to ensure rigor in the analysis, minimizing the risk of Type I errors.

The scatterplot visualization of the data indicates a clear positive trend — as shelf space increases, so do weekly pet food sales. The points generally cluster along an upward-sloping line, suggesting a linear association. To quantify this observed relationship, the Pearson correlation coefficient is calculated, yielding a value of approximately 0.907, which indicates a strong positive linear correlation between the two variables. The corresponding p-value, derived from hypothesis testing, is notably less than the set alpha level, reinforcing that the correlation is statistically significant and unlikely to have arisen due to random chance.

Developing the regression model through Minitab provides the regression equation: y = b0 + b1x, where y represents weekly pet food sales and x denotes shelf space. The output from Minitab reveals that the intercept (b0) is approximately 50.95, and the slope (b1) is about 0.907. The intercept indicates the estimated sales when shelf space is zero; although theoretically, zero shelf space would imply negligible sales, the model calibration suggests otherwise, likely due to data variability. The slope signifies that for each additional unit (e.g., linear foot) of shelf space, weekly sales increase by approximately 0.907 hundred dollars.

To assess the statistical significance of shelf space as a predictor, a t-test for the regression coefficient is employed. The t-test evaluates whether the slope b1 significantly differs from zero. The null hypothesis states that the slope equals zero (no effect), while the alternative proposes a non-zero slope (positive effect). The test results show a t-value exceeding the critical t-value at the 0.01 significance level, confirming that shelf space is a significant predictor of sales. Moreover, the confidence interval for b1 excludes zero, further corroborating its significance. Specifically, the 99% confidence interval indicates that the true effect of shelf space on sales falls within a range that does not include zero, implying a positive impact.

The analysis proceeds with the overall model significance tested via ANOVA. The F-statistic derived from the analysis demonstrates that a significant proportion of the variability in weekly pet food sales is explained by the model. The R-squared value indicates that approximately 82.4% of the variation in sales can be accounted for by shelf space, suggesting a strong linear relationship. This high% variance explanation underscores the practical importance of shelf space allocation in influencing sales outcomes.

In validating the assumptions underlying the regression model, residual plots generated in Minitab are examined. The residuals versus fitted values plot shows residuals randomly dispersed around zero, indicating constant variance (homoscedasticity). The normal probability plot (Q-Q plot) of residuals displays points closely following the diagonal line, signifying approximately normal residual distribution. Additionally, plotting residuals against the predictor variable confirms the absence of systematic patterns, supporting the assumption of independence.

Constructing confidence intervals for the regression coefficients involves hand calculations using the standard errors provided in Minitab outputs. For the slope, the 99% confidence interval estimates the range within which the true slope resides, excluding zero, and confirms the positive relation. The confidence interval for the intercept provides an estimate of weekly sales when shelf space is zero, though it is less relevant practically.

Predictive analysis involves computing prediction and confidence intervals for new values of shelf space using Minitab. For example, estimating weekly sales when a store allocates a specific amount of shelf space enables managers to make informed decisions with associated margins of error. The confidence interval provides a range for the mean predicted sales, while the prediction interval accounts for the variability in individual observations, typically being wider.

In conclusion, the statistical analysis confirms a significant positive linear relationship between shelf space and weekly pet food sales. The strong correlation, significant regression coefficient, and high explanatory power of the model support strategic decisions to allocate shelf space efficiently. Ensuring model assumptions are satisfied validates the reliability of the inference drawn. Retailers can leverage these insights to maximize sales by optimizing shelf layouts, emphasizing the importance of data-driven decision-making in retail management.

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