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MLC at Boise State 2018 M143Linear RegressionsGetting started with Making a Scatter Plot on your CalculatorSTEM Project Week #3Regression analysis is a method used to create a mathematical model to show therelationship between two quantities. These quantities are usually represented by variables.One of these variables is the dependent variable and the other is the independent variable.(Remember that an independent variable represents the value being changed, usually x,and the dependent variable is the variable that changes, usually y. y depends on x.)A regression equation is an equation that “best fits” a given set of data. It best describes therelationship between the two variables. The regression equation follows the trend of thedata.There are many types of equations that you will study and have studied in algebra. Someexamples of these equations are linear, quadratic, cubic, quartic, exponential, andlogarithmic. These are also the types of regression equations we can find.What is the purpose of finding regression equations? We find regression equations so wecan visualize the relationship between two quantities in order to make predictions or drawconclusions.In this activity, you will plot a set of data by hand, plot a set of data on your calculator as ascatterplot, find a regression equation using your graphing calculator, and analyze why youmay choose one type of regression equation over another. We will also examine domains,ranges, and functions.Getting Started With a Data SetThe Ideal Gas LawThe Ideal Gas Law is a thermodynamic equation of state for an ideal gas. An equation ofstate simply relates the state properties of a substance to the amount of that substance.Specifically, the ideal gas law relates the pressure (P), volume (V), and temperature (T) tothe amount (n) of an ideal gas.A one liter tank containing one mole of an ideal gas was heated tothe temperatures listed in row one. At each temperature, thepressure from a pressure gauge affixed to the tank was recordedin row two.T ( temp.Kelvin)300 310 320 330 340 350P (pressureatm)24.7 25.3 26.4 27 28 28.5Note: A “mole” is defined as thenumber of carbon atoms in 12grams of carbon-12, which isapproximately 6.022X10^23.MLC at Boise State 2018 M1432Plotting the Data by HandYou will be plotting the data on the grid shown below, graphing temperature on the x-axis(temperature is the domain and the independent variable). You will be graphing pressureon the y-axis (pressure is the range and the dependent variable). Recall that the domain isthe set of all possible values of an independent variable of a function and the range is theset of all possible values for a dependent variable of a function.Step 1: Label the x-axis and the y-axis of your grid. Let’s start with the x-axis. We will graphthe temperature on the x-axis.• What is your smallest x-value (temperature value)?• What should be your smallest x-value on your x-axis?(Make sure that your smallest value is on the x-axis. Youneed to have your x-axis go lower than the smallest value.)• What is your largest x-value (temperature value)?• What should be your largest x-value on your x-axis? (Makesure that your largest value is on the x-axis. You need tohave your x-axis go beyond the largest value.)• Label these values on your grid below.• What is your smallest y-value (pressure value)?• What should be your smallest y-value on your graph? (Make sure that your smallesty-value is on the y-axis. You need to have your y-axis go lower than the smallestvalue.)• What is your largest y-value (pressure value)?• What should be your largest y-value on your y-axis? (Make sure that your largestvalue is on the y-axis. You need to have your y-axis go beyond the largest value.)• Label these values on your grid below.Step 2: Now that your grid is labeled on the x- and y-axis, plot the data points from thetable onto the grid.In this excersice we will returnto the mathematical notation ofusing x for the horizontal axisand y for the vertical axis. In thefuture, we will use the variablelabels used in the problem andnot switch back to the standardmathematical x and y notation.MLC at Boise State 2018 M1433Step 3: Look at your data. If you were to find an equation of a function that best fits thisdata, would this equation be a line? Why or why not?Would this equation be a parabola? Why or why not?Would this equation be an exponential? Why or why not?**Make sure to remember that you are trying to find the equation that would “best fit” thegraph. Not the equation that exactly fits the graph.If you have plotted the data correctly, the equation that will best fit this data is a line. Use astraight edge to sketch a line that “best fits” the data. To sketch the line that best fits thedata, sometimes it is better to use a clear ruler. When sketching the line of best fit you aretrying to draw a line that comes as close to as many points as possible and balances theoverestimates and underestimates. In other words, the line should have the same numberof points above it as it does below it.Explain why your line best fits the data.Once you have drawn this line, you can write the equation by using any two points on it tofind the slope. Then use one of the points on the line, the slope you found and the pointslope equation of a line to write the equation.What is the equation of the line that best fits the data?MLC at Boise State 2018 M1434Finding the Equation for the Line of Best Fit Using Your CalculatorNow, you will enter this data into the calculator and use it to find the equation of your lineof best fit. We will compare this with the equation that you found above. Please use theinstructions on the document labeled Regression Instructions on the Math 143 MLCwebpage.First, find the set of instructions that says: “Graphing a set of data points on the TI-83”(making a scatterplot). Follow these instructions to plot the points on your calculator.Once you have plotted the data on your calculator and you see a plot similar to the one thatyou sketched, observe the scatterplot.Does your graph on the calculator look like the graph that you sketched? Why or why not?If they do not look the same you might want to explore reasons such as window size,domains and ranges, sketched points correctly or entered data correctly in the calculator.Once you have resolved any differences between your sketch and the calculator, look at thegraph on your calculator. Do you think that a line best fits the data? Why or why not?Do you think that an exponential equation best fits this data? Why or why not?Do you think that a quadratic best fits this data? Why or why not?If the data was entered and plotted correctly, it should most resemble a line just like inyour sketched graph. Now you want to find a line that is a good “fit.” In the plotted data weshowed one method to find this line. Using your calculator we will use another method tofind this line.MLC at Boise State 2018 M1435When you use your calculator to find this line, the line is sometimes called the “leastsquares”regression line. Follow the steps given in the regressions document labeled“finding the linear regression equation” to find the equation of this line.What is your equation of the line of best fit?In your calculator, you use x and y as your variables. Inchemistry, our variables are P for pressure and T fortemperature so rewrite your equation using P and Tinstead of x and y. What is your equation using P and T?Finally, you can use a linear regression to create amathematical model that represents the relationshipbetween two variables. Use your line of best fit to predictwhat the pressure will be when you raise the temperatureto 450 Kelvin. What will be the pressure when thetemperature is lowered to 250 Kelvin?ExercisesFor the data sets below:• determine the domain and range• plot the data on the graph making sure to label your “window” on your paper• sketch a line of best fit• enter the data into your calculator• find the line of best fit• compare your sketched line of fit to that of your calculators found line of best fitIn the Ideal Gas Law equation isP =nRTVwhereP = pressure of an enclosed group of gasmolecules in atmospheres (atm)V = volume of an enclosed group of gasmolecules in units of liters (L)n = moles of gas ( 1 mole of gas =6.022X10^23 molecules)T = Temperature of gas molecules in units ofKelvin (K)R = Gas law constant = 0.08206(L*atm)/(mol*Kelvin).In our situation we left n and V constant. Weleft both of them as 1. Thus the Ideal Gas Lawequation when V and n are 1 isP = RT .When we wrote our equation above, we left Pand T in our equation as our x and y. Thusyour above equation should look very closeto P = 0.08206T.MLC at Boise State 2018 M14361. A student is hanging masses from a spring and measuring the resulting stretch in thespring. The following table shows the students’ collected data.m (mass in grams) 15 25 35 45 55x (stretch in cm) 7.2 10.6 14.3 21.6 24.62. A student is hanging masses from a spring and measuring the resulting stretch in thespring. The following table shows the students’ collected data.m (mass in grams) 2 6 9 15 22x (stretch in cm) 15.3 21.0 25.1 33.4 43.2
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