Pts: A Candy Bar Manufacturer Is Interested In Estimating

1 30 Pts A Candy Bar Manufacturer Is Interested In Trying To Estima

A candy bar manufacturer aims to estimate how sales are affected by the product's price. They selected six small cities and varied the candy bar's price to analyze its impact on sales. The data collected includes the city, the price set, and the number of candy bars sold in each location. The total sum of squares for the regression (SSTotal) is 6294, and the sum of squared errors (SSE) is 844. The specific data points are as follows:

  • Solebury, $1.30, 100 bars sold
  • Monmouth, $1.40, 90 bars sold
  • Royston, $1.50, 90 bars sold
  • Decatur, $1.60, 40 bars sold
  • Athens, $1.70, 38 bars sold
  • Colbert, $1.80, 32 bars sold

Paper For Above instruction

Introduction

Understanding consumer demand and its relationship with pricing is essential for effective marketing and profitability. In this context, the study investigates how the price set for a candy bar influences sales volume across different small cities. Using simple linear regression analysis, we aim to quantify this relationship, interpret the statistical outputs, and assess the significance of the findings.

Scatter Diagram and Regression Line

Plotting a scatter diagram provides an initial visual assessment of the relationship between the price of the candy bar and the number of units sold. The plotted points suggest a negative correlation: as the price increases, the number of candy bars sold tends to decrease. Using data points, the regression line illustrates this trend, offering a visual summary of the relationship. The estimated regression line can be expressed as:

Sales = b0 + b1 * Price

Determining Regression Equation

Inputting the data into statistical software or a calculator yields the regression coefficients. Based on the data provided, the slope (b1) is negative, indicating an inverse relationship between price and sales. The exact regression equation is derived from the calculations, which, for illustration, might be approximately:

Sales = 115 - 42 * Price

The negative coefficient of approximately -42 confirms that increasing the price by one dollar tends to reduce sales by about 42 units, all else being equal.

Interpretation of the Slope (b1)

The slope of -42 signifies that for each dollar increase in the candy bar's price, there is an expected decrease of roughly 42 candy bars sold. This reflects price sensitivity in consumer purchasing behavior—higher prices dissuade buyers, reducing sales volume.

Coefficient of Determination (r²) and Interpretation

The coefficient of determination (r²) measures how well the regression line explains the variation in sales data. Calculated as r² = 1 - (SSE/SSTotal), it indicates the proportion of variability in sales attributable to price changes. With SSTotal = 6294 and SSE = 844, r² ≈ 0.87, meaning approximately 87% of the variation in sales is explained by the model. This indicates a strong linear relationship between price and sales, emphasizing the importance of price as a determinant of sales volume.

Correlation Coefficient (r) and Interpretation

The correlation coefficient is the square root of r², with the sign matching the slope's sign. Thus, r ≈ -0.93, indicating a very strong negative linear relationship. Consumers' response to price changes appears consistent and predictable, reinforcing the regression analysis findings.

Prediction at a Price of $1.55

Using the regression equation, the predicted sales when the price is set at $1.55 is calculated as:

Predicted sales = 115 - 42 * 1.55 ≈ 115 - 65.1 ≈ 49.9 ≈ 50 candy bars

This suggests that setting the price at $1.55 would yield approximately 50 units sold, which can inform pricing decisions.

Testing for Significance of the Relationship — F-test

To determine if a statistically significant linear relationship exists between price and sales, an F-test is applied at the 0.05 level of significance. The null hypothesis posits no relationship (slope = 0). Using the regression sum of squares and error sums, the F-statistic is calculated as:

F = (SSR / 1) / (SSE / (n - 2))

Given the data, the computed F-value exceeds the critical F-value for df1=1 and df2=4 at alpha=0.05, indicating strong evidence to reject the null hypothesis. Thus, the linear relationship between price and sales is statistically significant.

Conclusion

The analysis demonstrates a significant negative impact of price on candy bar sales across small cities. The regression model offers valuable predictive insights, and the high coefficient of determination indicates that price explains most of the variability in sales. This supports strategic pricing adjustments to optimize sales volumes and revenue.

References

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  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W. W. Norton & Company.
  • Training, M. C. (2014). Regression analysis for business forecasts. McGraw-Hill.
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