NUMB3RS Activity: How Does it Fit?

Transcription

1 Name Regression 1

2 NUMB3RS Activity: How Does it Fit? A series of sniper shootings has reduced the city of Los Angeles to a virtual ghost town. To help solve the shootings, the FBI has enlisted the help of Charlie Eppes as well as Special Agent Edgerton, a sniper instructor from Quantico. When Charlie becomes frustrated in his attempts to find a pattern in the data, Agent Edgerton suggests that there are factors that the equations cannot take into account. Later, the same sentiment is echoed by Charlie s friend Larry when Charlie comments that he has a pattern of patternlessness. How does Charlie know when the equations are a good fit to the data? To determine how well a line fits a set of data you will calculate the Correlation Coefficient, denoted commonly by the variable r. All values of the correlation coefficient are between -1 and 1. When r = -1 or 1, this represents data that are perfectly aligned with the line. The closer the value of r is to either -1 or 1, the better the data fits the equation. Look at these examples: Notice that when the correlation coefficient is positive, the slope of the line of best fit is positive, and when the correlation coefficient is negative, the slope of the line of best fit is negative. Also, you can notice that the closer the value is to 1 or -1 the better the line fits the data. When r = 0, this presents a special case. Although we might me tempted to believe that there is no pattern to our data when r = 0 (as in the graph on the left above), we can see from the other examples that there can still be a pattern to the data. That is, although the dispersion of data points is not linear, a pattern may still exist. 2

3 In the episode, Charlie is analyzing ballistic data where the x coordinate is the bullet weight in 100 grains and the y coordinate is the effective distance in 100 yards. Suppose he has collected the data below. ( 2,3,3,9,4,5,5,11,6,6,7,16,8,15 ) ( ) ( ) ( ) ( ) ( ) ( ) How well would a linear equation fit this data? Graph the values and draw a line of best fit. How well do you think the regression line fits the data? Explain your answer. Now let s find the value for the correlation coefficient using your TI-83 Plus/TI-84 Plus graphing calculator. Your calculator needs to be set up to display the diagnostic values for the regression models. Select DiagnosticOn from the Catalog menu. This only need to be done once so that the correlation coefficient is displayed on the calculator, and does not need to be repeated unless the calculator is reset to the defaults. Press 2 nd [CATALOG] to select the catalog. Scroll down to DiagnosticOn and press ENTER twice. 3

4 You will also need to use Stat Plots 1. The steps below outline how to check the settings for the stat plot. Press Y=. If there are previously entered equations, use the CLEAR button to erase them. Then press and press ENTER to toggle Plot1. If you wish to check the settings, press 2nd [STAT PLOT]. Press ENTER to select Plot1 and use the settings shown in the display above. You should also know how to clear lists used by stats. To view the stats table press STAT and select 1:Edit To clear a column, press the key and move the cursor over the name of the list you want to clear. Now, press CLEAR and then press. (Note: Do not press the DEL key; this will delete the list instead of clearing it.) Activity Press STAT and select Using the data from the Press STAT, go to the 1:Edit activity, enter the x-values CALC menu, and select in list L 1 and enter the 4:LinReg(ax+b) y-values in list L 2. 4

5 Next press VARS and go to Press ENTER. Your Press 2nd [STAT PLOT] and the Y-VARS menu. Select calculator will now choose Plot1. Use the 1:Function and then compute the values settings shown above; this choose 1:Y 1.This will now of a, b, and r. will create a scatterplot for calculate the equation of the data in lists L 1 and L 2. the regression line, and will store it as equation Y 1. Press ZOOM and select 9:ZoomStat to display the scatterplot of the data and the regression line. Notice how the graph of the line fits the scatterplot. Now answer the questions below. 1. What are the values calculated for a, b, and r to the nearest thousandth? 2. What is the equation of the line of best fit? 3. Were you surprised by the value of the correlation coefficient? 5

6 Practice / Homework Linear Regression 1. In a mathematics class of ten students, the teacher wanted to determine how a homework grade influenced a student s performance on the subsequent test. The homework grade and subsequent test grade for each student are given in the accompanying table. Homework Grade Test Grade Give the equation of the linear regression line for this set of data. Round the regression coefficients to the nearest hundredth. A new student comes to the class and earns a homework grade of 78. Based on the equation in part a, what grade would the teacher predict the student would receive on the subsequent test, to the nearest integer? 2. The accompanying table shows the enrollment of a preschool from 1980 through Write a linear regression equation to model the data in the table. Let t = 0 represent Year (t) Enrollment (y)

7 3. A factory is producing and stockpiling metal sheets to be shipped to an automobile manufacturing plant. The factory ships only when there is a minimum of 2,050 sheets in stock. The accompanying table shows the day, x, and the number of sheets in stock, f(x). Day Sheets in Stock Write the linear regression equation for this set of data, rounding the coefficients to four decimal places. Use this equation to determine the day the sheets will be shipped. 4. The data table below shows water temperatures at various depths in an ocean. Water Depth (x) (meters) Temperature (y) ( C ) Write the linear regression equation for this set of data, rounding all values to the nearest thousandth. Using this equation, predict the temperature (ºC), to the nearest integer, at a water depth of 255 meters. 7

8 Other types of Regression A model is a mathematical representation of the relationship between two real-world quantities. Exponential Regression y = ab x If a cup of coffee is left on a countertop, it will cool off slowly. The following table shows that temperature of a cup of coffee is sitting for 50 minutes. Time Temp Construct a scatter plot of the data. Find the exponential equation. Round values to the nearest thousandth. Using this equation, what should the temperature be, to the nearest hundredth, in 1 hour? Based on your equation, when, to the nearest minute, will the temperature be 80? 8

9 Power Regression y = ax b The accompanying table shows the number of new cases reported by the Nassau and Suffolk County Police Crime Stoppers program for the years 2000 through Year (x) New Cases (y) If x = 1 represents the year 2000, and y represents the number of new cases, find the equation of best fit using a power regression, rounding all values to the nearest thousandth. Using this equation, find the estimated number of new cases, to the nearest whole number, for the year Natural Logarithmic Function y = a + bln( x) The table below indicates the amount of sleep required for people at different ages. Age (yr.) Amount of Sleep Determine the natural logarithmic regression for this data. Round the regression coefficients to the nearest hundredth. Find to the nearest tenth of an hour the number of hours of sleep required by a 35-year-old. 9

10 Practice / Homework Other Types of Regression 1. The table below shows the results of an experiment involving the growth of bacteria. Time (x) in minutes # of Bacteria (y) Write a power regression equation for this set of data, rounding all values to three decimal places. Using this equation, predict the bacteria s growth, to the nearest integer, after 15 minutes. 2. The accompanying table shows wind speed and the corresponding wind chill factor when the air temperature is 10 F. Wind Speed (mi/h) Wind Chill Factor ( F ) Write the logarithmic regression equation for this set of data, rounding coefficients to the nearest ten thousandth. Using this equation, find the wind chill factor, to the nearest degree, when the wind speed is 50 miles per hour. Based on your equation, if the wind chill factor is 0, what is the wind speed, to the nearest mile per hour? 10

11 3. Jean invested $380 in stocks. Over the next 5 years, the value of her investment grew, as shown in the accompanying table. Years Since Investment (x) Value of Stock, in Dollars (y) Write the exponential regression equation for this set of data, rounding all values to two decimal places. Using this equation, find the value of the stock, to the nearest dollar, 10 years after her initial purchase. Determine the number of years it will take for the value of Jean s stocks to reach $ Donnie wanted to determine if the length of a pendulum has any relationship to the time required for the pendulum to complete one oscillation. He has placed the information in a table as shown below. Length (in feet) Time (in sec) Write an equation for a power function to model the data. Round values to the nearest thousandth. Use your formula to find the time, to the nearest tenth, required for a 5-foot pendulum to complete one oscillation. Find the length of the pendulum, to the nearest tenth, at 2.5 seconds. 11

12 5. A box containing 1,000 coins is shaken, and the coins are emptied onto a table. Only the coins that land heads up are returned to the box, and then the process is repeated. The accompanying table shows the number of trials and the number of coins returned to the box after each trial. Trial Coins Returned 1, Write an exponential regression equation, rounding the calculated values to the nearest ten-thousandth. Use the equation to predict how many coins would be returned to the box after the eighth trial. 6. Water is draining from a tank maintained by the Yorkville Fire Department. Students measured the depth of the water in 15-second intervals and recorded the results in the accompanying table. Time (in seconds) Depth of Water (in feet) Write the power regression for this set of data, rounding all values to the nearest ten thousandth. Using this equation, predict the depth of the water at 2 minutes, to the nearest tenth of a foot. 12