A+ Work

Curve itting Project Linear Model Instructions

For this assignment, collect data exhibiting a relatively linear trend, find the line of best fit, plot the data and the line, interpret the slope, and use the linear equation to make a prediction. Also, find r2 (coefficient of determination) and r (correlation coefficient). Discuss your findings. Your topic may be that is related to sports, your work, a hobby, or something you find interesting. If you choose, you may use the suggestions described below. 

A Linear Model Example and Technology Tips are provided in separate documents.

Tasks for Linear Regression Model (LR)

(LR-1) Describe your topic, provide your data, and cite your source. Collect at least 8 data points. Label appropriately.  (Highly recommended: Post this information in the Linear Model Project discussion as well as in your completed project. Include a brief informative description in the title of your posting. Each student must use different data.) 
 
The idea with the discussion posting is two-fold: (1) To share your interesting project idea with your classmates, and (2) To give me a chance to give you a brief thumbs-up or thumbs-down about your proposed topic and data. Sometimes students get off on the wrong foot or misunderstand the intent of the project, and your posting provides an opportunity for some feedback. Remark: Students may choose similar topics, but must have different data sets. For example, several students may be interested in a particular Olympic sport, and that is fine, but they must collect different data, perhaps from different events or different gender.

(LR-2) Plot the points (x, y) to obtain a scatterplot. Use an appropriate scale on the horizontal and vertical axes and be sure to label carefully. Visually judge whether the data points exhibit a relatively linear trend. (If so, proceed. If not, try a different topic or data set.)

(LR-3) Find the line of best fit (regression line) and graph it on the scatterplot. State the equation of the line.

(LR-4) State the slope of the line of best fit. Carefully interpret the meaning of the slope in a sentence or two.

(LR-5) Find and state the value of r2, the coefficient of determination, and r, the correlation coefficient. Discuss your findings in a few sentences. Is r positive or negative? Why? Is a line a good curve to fit to this data? Why or why not? Is the linear relationship very strong, moderately strong, weak, or nonexistent?

(LR-6) Choose a value of interest and use the line of best fit to make an estimate or prediction. Show calculation work.

(LR-7) Write a brief narrative of a paragraph or two. Summarize your findings and be sure to mention any aspect of the linear model project (topic, data, scatterplot, line, r, or estimate, etc.) that you found particularly important or interesting. 

You may submit all of your project in one document or a combination of documents, which may consist of word processing documents or spreadsheets or scanned handwritten work, provided it is clearly labeled where each task can be found. Be sure to include your name. Projects are graded on the basis of completeness, correctness, ease in locating all of the checklist items, and strength of the narrative portions.

For this assignment you will implement a project involving nonlinear curve-fitting and interpretation. You will assess the appropriateness of nonlinear models, and explore the predictive power of the models. You will use appropriate technology to perform the modeling tasks.

There are instructions below for two models: a quadratic model and an exponential model.

The nonlinear model project is due on May 1.

Quadratic Regression (QR)

Data: On a particular day in April, the outdoor temperature was recorded at 8 times of the day, and the following table was compiled.

REMARKS: The times are the hours since midnight. For instance, 7 means 7 am, and 13 means 1 pm.

The temperature is low in the morning, reaches a peak in the afternoon, and then decreases.

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. (r2 is calculated using a different formula than for linear regression. However, just as in the linear case, the closer r2 is to 1, the better the fit. Just work with r2, not r.) Is a parabola a good curve to fit to this data?

(QR-4) Use the quadratic polynomial to make an outdoor temperature estimate. Each class member will compute a temperature estimate for some time of day.  Please choose any time of day you like in order to make this estimate. Be sure to use the quadratic regression model to make the estimate (not the values in the data table). State your results clearly -- the time of day and the corresponding outdoor temperature estimate.

(QR-5) Using algebraic techniques we have learned, find the maximum temperature predicted by the quadratic model and find the time when it occurred. Report the time to the nearest quarter hour (i.e., __:00 or __:15 or __:30 or __:45). (For instance, a time of 18.25 hours is reported as 6:15 pm.) Report the maximum temperature to the nearest tenth of a degree. Show work.

(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.  Please choose a suitable specific target temperature. Then, report the time(s) to the nearest quarter hour.  Be sure to use the quadratic model to make the time estimates (not values in the data table). Show work. State your results clearly -- the target temperature and the associated time(s). Show work.

Please see the Technology Tips topic for additional information about generating the scatterplot and quadratic polynomial.

Exponential Regression (ER)

Data: A cup of hot coffee was placed in a room maintained at a constant temperature of 69 degrees. 

The temperature of the coffee was recorded periodically, and the following table was compiled.

Table 1:

REMARKS: Common sense tells us that the coffee will be cooling off and its temperature will decrease and approach the ambient temperature of the room, 69 degrees.

So, the temperature difference between the coffee temperature and the room temperature will decrease to 0.

 We will be fitting the data to an exponential curve of the form y = A e- bx. Notice that as x gets large, y will get closer and closer to 0, which is what the temperature difference will do.

 So, we want to analyze the data where x = time elapsed and y = T - 69, the temperature difference between the coffee temperature and the room temperature.

Table 2

Tasks for Exponential Regression Model (ER)

(ER-1) Plot the points (x, y) in the second table (Table 2) 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.

(ER-2) Find the exponential function of best fit and graph it on the scatterplot. State the formula for the exponential function. It should have the form y = A e- bx where software has provided you with the numerical values for A and b.

(ER-3) Find and state the value of r2, the coefficient of determination. Discuss your findings.(r2 is calculated using a different formula than for linear regression. However, just as in the linear case, the closer r2 is to 1, the better the fit.) Is an exponential curve a good curve to fit to this data?

(ER-4) Use the exponential function to make a coffee temperature estimate. Please choose a suitable temperature estimate. Substitute your x value into your exponential function to get y, the corresponding temperature difference between the coffee temperature and the room temperature. Since y = T - 69, we have coffee temperature T = y + 69. Take your y estimate and add 69 degrees to get the coffee temperature estimate. State your results clearly -- the elapsed time and the corresponding estimate of the coffee temperature.

(ER-5) Use the exponential function together with algebra to estimate the elapsed time when the coffee arrived at a particular target temperature.  Report the elapsed time to the nearest tenth of a minute. Please choose a target coffee temperature T of your own.

Given your target temperature T,  then y = T - 69 is your target temperature difference between the coffee and room temperatures. Use your exponential model y = A e-bx. Substitute your target temperature difference for y and solve the equation y = A e-bx for elapsed time x. Show algebraic work in solving your equation. State your results clearly -- your target temperature and the estimated elapsed time, to the nearest tenth of a minute.

For instance, if the target coffee temperature T = 150 degrees, then y = 150 - 69 = 81 degrees  is the temperature difference between the coffee and the room, what we are calling y. So, for this particular target coffee temperature of 150 degrees, the goal is finding how long it took for the temperature difference y to arrive at 81 degrees; that is, solving the equation 81 = A e- bx for x. 

Please see the Technology Tips topic for additional information about generating the scatterplot and exponential function