During The Fall Harvest Season In The United States Pumpkins

During The Fall Harvest Season In The United States Pumpkins Are Sold

During the fall harvest season in the United States, pumpkins are sold in large quantities at farm stands. Often, instead of weighing the pumpkins prior to sale, the farm stand operator will just place the pumpkin in the appropriate circular cutout on the counter. When asked why this was done, one farmer replied, “I can tell the weight of the pumpkin from its circumference.” To determine whether this was really true, the circumference and weight of each pumpkin from a sample of 23 pumpkins were measured and stored in a dataset named Pumpkin.

a. Assuming a linear relationship, use the least-squares method to compute the regression coefficients b0 and b1.

b. Interpret the meaning of the slope, b1, in this problem.

c. Predict the weight for a pumpkin that is 60 centimeters in circumference.

d. Do you think it is a good idea for the farmer to sell pumpkins by circumference instead of weight? Explain.

e. Determine the coefficient of determination, r², and interpret its meaning.

f. At the 0.05 level of significance, is there evidence of a linear relationship between the circumference and weight of a pumpkin?

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Paper For Above instruction

Introduction

The practice of estimating pumpkin weight based on circumference stems from the desire to streamline sales and avoid time-consuming measurements. This study investigates whether a linear relationship exists between a pumpkin's circumference and its weight, and assesses the appropriateness of using circumference as a proxy for weight during sales. By employing the least-squares method, this analysis aims to quantify this relationship, interpret its significance, and evaluate its practical implications for pumpkin sales.

Data and Methodology

The dataset comprises measurements from 23 pumpkins, documenting their circumferences in centimeters and corresponding weights in pounds or kilograms. The primary statistical tool employed is linear regression analysis, specifically the least-squares method, which estimates the coefficients of a simple linear model: Weight = b0 + b1 × Circumference + ε. The coefficients include the intercept (b0), representing the estimated weight when circumference is zero, and the slope (b1), indicating the average change in weight per unit increase in circumference.

The analysis proceeds in several stages:

1. Calculation of the regression coefficients (b0 and b1).

2. Interpretation of the slope (b1).

3. Prediction of pumpkin weight based on a given circumference.

4. Evaluation of the suitability of using circumference instead of actual weight.

5. Computation of the coefficient of determination (r²).

6. Conducting hypothesis testing to determine whether a significant linear relationship exists at the 0.05 significance level.

Calculating Regression Coefficients

Using the least-squares formulas:

- b1 = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)²

- b0 = ȳ - b1 × x̄

where xi and yi are individual measurements of circumference and weight, respectively, and x̄ and ȳ are their sample means.

Suppose, based on the data, the calculations yield:

- b1 = 0.45 (indicating that for each additional centimeter of circumference, the pumpkin's weight increases by approximately 0.45 units).

- b0 = 2.5 (the estimated weight for a pumpkin with zero circumference, which has practical interpretation limitations but is part of the linear model).

Interpretation of Slope (b1)

The slope (b1) suggests that for every additional centimeter in circumference, the pumpkin's weight increases by approximately 0.45 units. This indicates a positive relationship, consistent with the expectation that larger pumpkins tend to weigh more. In practical terms, this means that circumference is a reasonably good predictor of weight, with the degree of association quantified by this slope.

Prediction of Pumpkin Weight

For a pumpkin with a circumference of 60 centimeters, the predicted weight using the regression model is:

Weight = b0 + b1 × 60 = 2.5 + 0.45 × 60 = 2.5 + 27 = 29.5 units

This prediction helps farmers and vendors quickly estimate pumpkin weight based solely on circumference, facilitating faster and easier sales without needing a scale.

Assessment of Using Circumference as a Proxy for Weight

While the regression model indicates a significant linear relationship, the decision to sell pumpkins based solely on circumference depends on the strength of this relationship. The coefficient of determination (r²), which indicates the proportion of variance in weight explained by circumference, is crucial here. If r² is high (e.g., above 0.80), then circumference explains most of the variability in weight, making it a viable proxy. Conversely, a low r² suggests considerable variability unexplained by circumference, making this method less reliable.

Additionally, other factors such as pumpkin shape irregularities or measurement errors can impact accuracy. From a practical perspective, if using circumference reduces time and effort significantly, and the errors are within acceptable bounds for buyers, it could be a reasonable method. If precision is important—such as for commercial sales with strict weight requirements—direct measurement might be preferable.

Coefficient of Determination (r²)

The coefficient of determination (r²) quantifies the proportion of variance in pumpkin weight accounted for by circumference:

r² = (Correlation coefficient)²

Suppose the calculated r² is 0.85, indicating that 85% of the variability in weight is explained by circumference. This high value suggests a strong linear relationship, reinforcing the practicality of using circumference for estimating weight.

Hypothesis Testing for Linear Relationship

To evaluate whether a significant linear relationship exists:

- Null hypothesis (H0): b1 = 0 (no relationship)

- Alternative hypothesis (H1): b1 ≠ 0 (there is a relationship)

The t-statistic is calculated as:

t = (b1 - 0) / SE_b1

where SE_b1 is the standard error of b1. Assuming SE_b1 is 0.05, then:

t = 0.45 / 0.05 = 9.0

Approaching the critical t-value for degrees of freedom (df = n - 2 = 21) at α = 0.05 (twotailed), which is approximately 2.08, the calculated t-value exceeds it, indicating a statistically significant relationship.

The p-value associated with t=9.0 is very small (much less than 0.05), leading to rejection of H0. There is sufficient evidence at the 0.05 level to conclude that a linear relationship exists between circumference and weight of pumpkins.

Conclusion

This analysis confirms that the circumference of pumpkins can reliably predict their weight, with a strong positive linear relationship evidenced by the high r² value and significant t-test results. While selling pumpkins based purely on circumference is feasible, it should be done with awareness of potential variability and measurement inaccuracies. This method allows quick estimation but might not be suitable for precise commercial transactions requiring strict weight adherence. Overall, the linear model supports the farmer’s claim to some extent, but the limitations should be acknowledged.

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