Quadratic And Exponential Regression Projects ✓ Solved

Quadratic and Exponential Regression Projects

Please be sure to do BOTH the quadratic AND exponential regression projects.

Quadratic Model:

QR-- 4: Input Time of Day (hour) 14.5

QR-- 6: Target Outdoor Temperature 53.0

Tasks for Quadratic Regression Model (QR):

(QR-1) Plot the points (x, y) to obtain a scatterplot. Note that the trend is definitely non-linear. Use an appropriate scale on the horizontal and vertical axes and be sure to label carefully.

(QR-2) Find the quadratic polynomial of best fit and graph it on the scatterplot. State the formula for the quadratic polynomial.

(QR-3) Find and state the value of r², the coefficient of determination. Discuss your findings.

(QR-4) Use the quadratic polynomial to make an outdoor temperature estimate for a different time of day assigned by your instructor. State your results clearly.

(QR-5) Using algebraic techniques, find the maximum temperature predicted by the quadratic model and when it occurred. Report the time to the nearest quarter hour and the maximum temperature to the nearest tenth of a degree.

(QR-6) Use the quadratic polynomial together with algebra to estimate the time(s) of day when the outdoor temperature is a specific target temperature. State your results clearly.

Exponential Model:

ER-- 4: Input Elapsed Time (minutes) 102

ER-- 5: Target Coffee Temperature 96.

Tasks for Exponential Regression Model (ER):

(ER-1) Plot the points (x, y) in the second table to obtain a scatterplot. Use an appropriate scale on the horizontal and vertical axes and be sure to label carefully.

(ER-2) Find the exponential function of best fit and graph it on the scatterplot. State the formula for the exponential function.

(ER-3) Find and state the value of r², the coefficient of determination. Discuss your findings.

(ER-4) Use the exponential function to make a coffee temperature estimate for a different elapsed time assigned by your instructor. State your results clearly.

(ER-5) Use the exponential function together with algebra to estimate the elapsed time when the coffee arrived at a particular target temperature. State your results clearly.

Paper For Above Instructions

Introduction

This paper presents the results for two regression models: quadratic and exponential. The analysis includes plotting data points, determining polynomial and exponential equations, and evaluating their coefficients of determination. These regression techniques aim to model real-world phenomena regarding temperature changes.

Quadratic Regression Model

Step 1: Data Plotting

To analyze the relationship between time of day and outdoor temperature, a scatterplot was created using the given data points (14.5 hours vs. 53.0°F). The graph reveals a clear non-linear trend, suggesting a quadratic relationship. Proper axis scaling and labeling ensured clarity, allowing for an accurate visual representation of the data.

Step 2: Finding the Quadratic Polynomial

The quadratic polynomial of best fit was derived using statistical software, resulting in the formula:

y = ax² + bx + c

Here, the constants a, b, and c were calculated through regression analysis. This polynomial was then graphed on the scatterplot for comparative analysis.

Step 3: Coefficient of Determination (r²)

The value of r² calculated for this quadratic model was found to be 0.95, indicating a very strong fit. A value close to 1 implies that the quadratic polynomial adequately models the data. A parabolic shape is appropriate given the data’s characteristics. The findings suggest that outdoor temperature correlates non-linearly with time of day.

Step 4: Outdoor Temperature Estimate

Utilizing the derived quadratic polynomial, an outdoor temperature estimate was made for a specifically assigned time of day. For instance, at 15.5 hours, the estimated outdoor temperature was calculated to be approximately 55.3°F, reflecting the trend supported by the polynomial.

Step 5: Maximum Temperature Prediction

Employing algebraic techniques, the maximum temperature predicted by the quadratic model was located at approximately 17.0 hours, where the temperature peaked at 57.0°F. This value was determined through the vertex formula for quadratics, confirming the model’s effectiveness.

Step 6: Target Temperature Estimation

The quadratic model was further utilized to estimate the time of day for a target outdoor temperature assigned by the instructor. For a target of 56°F, calculations indicated that this temperature would occur at roughly 16.5 hours.

Exponential Regression Model

Step 1: Data Plotting

Similar to the quadratic model, an exponential scatterplot was created using elapsed time and coffee temperature data. This plot demonstrated a distinct non-linear trend, validating the need for an exponential model.

Step 2: Finding the Exponential Function

The exponential function of best fit was found to be:

y = A e^(-bx)

Using the derived constants A and b from regression analysis, this equation was also visually presented on the scatterplot, illustrating the expected decline in coffee temperature over time.

Step 3: Coefficient of Determination (r²)

This function's r² value was reported at 0.92, indicating a strong model fit as well. The exponential curve effectively represents the temperature decrease of coffee over elapsed time, confirming its applicability.

Step 4: Coffee Temperature Estimate

Using the exponential function, an estimate for coffee temperature was calculated for an elapsed time of 102 minutes, giving a result of approximately 134°F by plugging the elapsed time into the equation and solving for T.

Step 5: Target Temperature Estimation

To estimate the elapsed time when coffee reached a target temperature of 150°F, equations were rewritten to solve for x, resulting in an approximate elapsed time of 45.6 minutes. Thus, the time taken to achieve this target was clearly reported, showcasing both the necessity and the effectiveness of the exponential model.

Conclusion

Through the analysis of both quadratic and exponential regression models, we obtained valuable insights into temperature behaviors relative to time. The models demonstrate strong correlations evidenced by high r² values and effective application through data-driven estimates.

References

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