S9 - Statistics with Graphing Calculators

Transcription

1 Summer 2006 I2T2 Probability & Statistics Page 165 S9 - Statistics with Graphing Calculators Exploring the different graphs possible: NY Standards: 7.S.4, 5, 6; A.S.4, 5, 6, 7, 8, 14; A2.S.6 1. Enter data: Go to STAT Edit. To enter the column labels, to L1, press 2nd INS to insert a new column. Enter ST for Student, then Enter. Enter the numbers 1, 2, 3, 4, 5, 6, 7, 8 for the students. Student Verbal Score Math Score Angella Bob Carol David Jean Lisa Peter Matthew Similarly insert and label the column for Verbal Scores VS and the column for Math Scores MS. Enter the SAT scores from the chart above. Press 2nd QUIT to return to the home screen. 3. Statistical computations: Go to STAT, CALC, select 1-Var Stats, press Enter to the home screen.. Then 2nd LIST to VS, Enter, Enter. These same computations are available individually under 2nd STAT MATH. then to mean(, stddev(, median(, min(, max(.

2 Summer 2006 I2T2 Probability & Statistics Page 166 Record the statistics in the table below. Repeat for the Math Score. Record the Mean: Record the Standard Deviation: Record the Median: Record the Minimum: Record the Maximum: Calculate the Range: Verbal Score Math Score 4. Box Plots: First press Y= and clear out any functions there. Then go to 2nd STAT PLOT, select Plot 1, Enter. Press Enter to turn Plot 1 On, to Type, then to the box plot icon (also called box and whiskers), Enter. Then to Xlist. Go to 2nd LIST to VS, Enter. Press ZOOM, then to ZoomStat. You now see the box plot for the verbal scores. Repeat the above steps, using Plot 2, for the Math Scores. Go to ZoomStat again to graph the two box plots together. (It has to resize the window for the second plot.) What statistics are used to create the box plots? Use TRACE to help determine this. Use the arrows to go between plots. The "whiskers" are the long lines extending from each box. A point that is at an extreme, where the distance between that point and the median is greater than 1.5 times the distance between Q1 and Q3, is called an "outlier." Looking at the data, do you think that one of the whiskers is exceptionally long due to an outlier? You might have noticed in the STAT PLOT menu there were two box plots. The one with a dot at the end of one whisker does not connect outliers, but shows them as separate points.

3 Summer 2006 I2T2 Probability & Statistics Page 167 Shows no outliers. 5. Calculations with the Data: Return to the statistics editor. (STAT EDIT). Make a new list called CS for combined scores. Position the cursor on the new label and press ENTER. You should have a blinking cursor at the bottom after the name. Enter the following: Go to 2nd LIST to MS, Enter. Then type +, then go to 2nd LIST to VS, Enter, Enter. The CS column now has a list of the sums of each score. If you arrow up to the name of the list, you will see {946, 1140, 1047, }. If you change anyone's score in MS or VS, CS will have to be recalculated because it contains a list, not the formula. To attach a formula to CS, after CS= type a " quotation mark, then follow the above instructions to enter the formula. When you press Enter, the same values appear in the column, but when you arrow up to the name, you will see the formula instead of the list. Now if you change anyone's score in MS or VS, the value in CS changes automatically. What is Angella's (1) combined score? Who has the highest combined score? 6. Histograms: Turn all the plots off by going to the STAT PLOT menu and choosing 4: PlotsOff. Press ENTER until you see Done on the home screen. Now go back to the STAT PLOT (or PLOT) menu and choose Plot 1 again. Turn Plot 1 on, choose the histogram. First graph the Verbal Scores, then the Math Scores, then the Combined Scores. Remember to use ZoomStat to get a good window. (Note that you cannot graph more than one histogram at a time with the graphing calculator.)

4 Summer 2006 I2T2 Probability & Statistics Page 168 To increase the number of columns, go back to WINDOW and change the Xscl to 10. Trace to see the range of each column and the number of students in that range. How might a school guidance counselor use this graph? 7. Sorting: To display the combined scores in ascending order, remember that we have to include the other three lists too to keep the correct scores with the correct person. Go to STAT, or go to 2nd LIST, OPS. Choose SortA(, then go to 2nd LIST to CS, Enter, comma, then go to 2nd LIST to ST, Enter, comma, then go to 2nd LIST to VS, Enter, comma, then go to 2nd LIST to MS, Enter, ) Enter. Now go back to STAT Edit. What are the benefits of sorting? How is this useful? Who has the highest combined score? 8. Scatter Plot: A scatter plot can give information on whether the two scores are related. For example, if the points seem to go from lower left to upper right, we consider the scores to have a positive correlation. In other words, there is some relationship between getting high math scores and high verbal scores. On the other hand, if the general trend is from upper left to lower right, there is a negative correlation. This would suggest that students, who score high in math, score low in verbal, and vice versa. Do you believe there should be a positive, negative, or no correlation between math and verbal SAT? We can test this conjecture with our small sample.

5 Summer 2006 I2T2 Probability & Statistics Page 169 Go to STAT PLOT, turn off all other plots except for Plot 1. Type should be scatter plot (dots), Xlist: VS, Ylist: MS, any Mark you prefer. Press ZOOM, ZoomStat and look at the graph. Is there a correlation? Does it make a difference if Xlist is MS and Ylist is VS? Switch them and regraph. 9. Determine the regression line: Go to STAT CALC, choose LinReg(ax+b), Enter. Now whatever is currently your Xlist (2nd LIST to MS), Enter, comma, whatever is currently your Ylist (2nd LIST to VS), Enter, comma, VARS, Y-VARS, Function, Y1, Enter, Enter. On the home screen, you should see the calculated values for a and b, and the correlation coefficient r. If you do not see a value for r and r 2, then do the following: Press 2nd CATALOG (above 0 on TI-84.) Arrow down to DiagnosticOn, press Enter. Then press 2nd ENTRY twice to get back to the LinReg command, and press Enter. The closer r is to +1 or 1, the better the fit between the function and the data. The r-value for this function tells you that a line is not a particularly good match for this data. Now press GRAPH to see the regression line graphed with your data. Because the regression line has a positive slope, you can see that your data has a positive correlation.

6 Summer 2006 I2T2 Probability & Statistics Page 170 Problem #9 The World's Fastest Men & Women NY Standards: 7.S.4, 6, 8, 12; A.S.7, 8, 17; A2.S.6, 7 Is there a pattern to the times recorded for the gold medallists in the men's 100-meter run from past Olympics? If there is a pattern, can you determine what the winning time in Barcelona, 1992 might have been? Don't look up the winning time yet! What about Atlanta in 1996, Australia in 2000, and Greece in 2004? A scatterplot is a good way to see if there is a pattern in the relationship between the year of the Olympics and the gold medal winning time. This scatterplot is easily created by hand, with computer software, or with graphing calculators. Before getting started with the graphing calculator, check the MODE settings and put them all on the default settings (the ones on the left). Entering data: First clear all lists by pressing 2nd MEM ClrAllLists. Now enter the data (STAT EDIT). Under L 1, enter the year of each entry and press ENTER after each one. Then go to the top of the List L 2 and enter the time for each year in the same order. When finished press 2nd QUIT to return to the home screen. Men s 100 Meter Run Gold Medallists for the Modern Olympic Games Year Name Time (Sec.) 1896 Thomas Burke, United States Francis W. Jarvis, United States Archie Hahn, United States Reginald Walker, South Africa Ralph Craig, United States Charles Paddock, United States Harold Abrahams, Great Britain Percy Williams, Canada Eddie Tolan, United States Jessie Owens, United States Harrison Dillard, United States Lindy Remigino, United States Bobby Morrow, United States Armin Hary, Germany Bob Hayes, United States Jim Hines, United States Valeri Borzov, USSR Hasely Crawford, Trinidad Allan Wells, Great Britain Carl Lewis, United States Carl Lewis, United States 9 99 Setting the window: Before drawing the scatterplot, set the WINDOW because the scatterplot is drawn on the same graphics screen you have used to graph functions. Our X data range from 1896 to 1988, so set Xmin to 1890 and Xmax to 2000 (or similar figures) and Xscl to 20. Because the winning times range from 12 down to 9.99, set Ymin to 9 and Ymax to 13 and Yscl to 0.5. Drawing the scatterplot: First clear all functions out of the Y= screen. Next go to STATPLOT by pressing 2nd, then Y=. At Plot 1, press ENTER. Under Plot 1, choose ON, then for TYPE choose the first icon for scatterplot. Next choose L 1 for Xlist and L 2 for Ylist (unless you named them something else). 2nd 1 for L1, 2nd 2 for L2.

7 Summer 2006 I2T2 Probability & Statistics Page 171 In order to see your points better, choose the first or second icon under Mark. Then press the GRAPH key. Finding the line of best fit: The TI-84 can determine very quickly the regression line for these data. From the Home screen (2nd QUIT), press STAT. Then under CALC, choose LinReg(ax +b). L1 and L2 are the default lists, so you do not have to enter anything unless you named the lists something else. To graph the line of best fit, we need to enter it into Y1. To do this, while still on the home screen, before pressing Enter after LinReg(ax+b), do the following: Press VARS, then arrow to Y-VARS, choose Function, then choose Y1. Press ENTER and you will see values for a and b. Record these values below. a = b = The value of b represents the y-intercept, a represents the slope of the regression line. Now press GRAPH and you will have the scatterplot and the line of best fit. What part of the data does b represent? How can we interpret the value of a with respect to the data? Is a increasing or decreasing? What does that mean with respect to running times? With this mathematical model you can now make a number of predictions about Olympic times. 1. What might have been a good winning time in Barcelona? There are several ways to answer that question. One is to trace the function, get X as close to 1992 as you can, and read Y. Another is to store 1992 into X, go to VARS, Y- VARS, choose Y1, and press Enter. A third method is to use the TABLE menu. 2. Now go to a reference book or the Internet. Who won in Barcelona? What was the actual winning time? 3. What time might have been expected for the winner at the Atlanta Games in 1996? 4. What was the actual winning time? 5. Notice from the table on the previous page that there is no time listed for some years. Why not? Use your mathematical model to predict what the winning times might have been, had there been Olympics in those years. 6. Using the model, what should we have expected for Australia in 2000? What was the actual winning time? 7. What should we expect for 2004 in Greece? What was the actual winning time?

8 Summer 2006 I2T2 Probability & Statistics Page 172 Fat Versus Calories NY Standards: 7.S.4, 6, 8; A.S.7, 8, 13, 14, 17; A2.S.6, 7 The following table contains the number of calories and grams of fat for selected fast foods. Does the scatterplot of these variables for each item look linear? Is there a positive correlation between fat and calories the more fat, the higher the number of calories? Fast food item Grams of fat Calories Burger King Whopper McDonald's Big Mac Wendy's Big Classic Arby's Roast Beef Hardee's Roast Beef Roy Roger's Roast Beef Burger King Whaler McDonald's Filet-O-Fish Arby's Chicken Breast Sandwich Burger King Chicken Tenders Church's Fried Chicken (2 pc.) Hardee's Chicken Filet Sandwich Kentuckv Fried Chicken (2 Pc.) Kentucky Fried Chicken Nuggets McDonald's Chicken Nuggets Roy Roger's Chicken (2 pc.) Wendy's Chicken Fillet Sandwich To Graph, remember to do the following: Clear all graphs in the Y= screen and turn off all other STAT PLOTS. Enter the data as sets of ordered pairs with x as the number of grams of fat and y as the number of calories. Select an appropriate viewing window, such as [10,40] by [200,600]. Choose Xscl and Yscl so you don't have more than 20 hash marks on each. Choose LinReg(ax +b) and put your equation in Y 1 or Y2. 1. The standard form of the regression line uses a as slope and b as the y-intercept, so an approximate equation of a line through the data is: y = 2. What kind of correlation is there between fat and calories positive, negative, or neither? 3. What does b, the y-intercept, mean in regards to fat and calories? What does the value of a tell us regarding fat vs calories?

9 Summer 2006 I2T2 Probability & Statistics Page If you found out that a particular meal at Perkins had 575 calories, what would you expect the grams of fat to be? 5. If you found out that a Steak & Shake hamburger contained 15 grams of fat, what would you expect the calories to be? Scrabble statistics. NY Standards: 7.S.4, 6, 8, 12; A.S.7, 8, 17; A2.S.6, 7 This problem is based on an activity, A-B-C, from the Mathematics Teacher, April 2004, by Mary Richardson and John Gabrosek. You can read about the history of the game on the website or In the activity, students are to emulate how Alfred Butts determined the number of each letter tile by tallying the number of times each letter of the alphabet appears in an article. The purpose of the activity is to examine the relationship between a letter s relative frequency in an article, the percent of Scrabble tiles per letter, and the point value of each tile. Our variation will just look at the relationship between the frequency of each letter in a Scrabble game and its point value. There are 100 tiles in a Standard Edition designed for American English. In a classroom, you could have students count and record the frequency of each tile and its point value. I was able to locate the following information on the Internet. Notice that no point values were given for N and C. Letter Value Distribution Letter Value Distribution Letter Value Distribution A 1 9 J 8 1 S 1 4 B 3 2 K 5 1 T 1 6 C? 2 L 1 4 U 1 4 D 2 4 M 3 2 V 4 2 E 1 12 N? 6 W 4 2 F 4 2 O 1 8 X 8 1 G 2 3 P 3 2 Y 4 2 H 4 2 Q 10 1 Z 10 1 I 1 9 R 1 6 blank Do you think the association between the Scrabble tile value and the letter s relative frequency will be positive or negative? Why? 2. Using lists and Plot 1, make a scatterplot with each letter s Scrabble value on the vertical axis and each letter s distribution on the horizontal axis. (Do not use the numbers for the blank.) 3. Using the scatterplot, describe the form, strength, and direction of the association between a letter s distribution and its value.

10 Summer 2006 I2T2 Probability & Statistics Page Use the graphing calculator to fit a straight-line model to the data. (The regression equations model average point values and may predict values that are not possible for actual data.) 5. Turn off Plot 1 and the line equation. Use your calculator to compute the residuals for the straight-line model. Using Plot 2, plot the residuals (on the vertical axis) versus the distribution of each letter (on the horizontal axis). Change your window to accommodate both positive and negative y-values. Give an intuitive explanation for the pattern of this plot. Whenever the graph of the residuals follow a pattern, are not randomly scattered, then this is an indication that the model chosen is not a good model. Based on your observation of the residual graph, do you think a straight-line model is a good model to use to describe the relationship between a letter s distribution and its Scrabble tile point value? 6. Turn off Plot 2. Use your calculator to fit the quadratic model y = a! x 2 + b! x + c to the data. Turn on Plot 1 to graph your fitted quadratic equation on the scatterplot produced in question Turn off Plot 1 and the quadratic graph. Use your calculator to compute the residuals for the quadratic model. Turn on Plot 2 to graph the residuals (on the vertical axis) versus the distribution of each letter (on the horizontal axis). Change your window to accommodate both positive and negative y-values. Give an intuitive explanation for the pattern of this plot. Whenever the graph of the residuals follow a pattern, are not randomly scattered, then this is an indication that the model chosen is not a good model. Based on your observation of the residual graph, do you think a quadratic model is a good model to use to describe the relationship between a letter s distribution and its Scrabble tile point value? 8. Turn off Plot 2. Use your calculator to fit the cubic model y = a! x 3 + b! x 2 + c! x + d to the data. Turn on Plot 1 to graph your fitted cubic equation on the scatterplot produced in question Repeat Step 7 with the residuals. Based on your observation of the residual graph, do you think a cubic model is a good model to use to describe the relationship between a letter s distribution and its Scrabble tile point value? 10. Use your cubic regression equation to predict the Scrabble-tile point value for the letters C and N. (Actual values are C = 3, N = 1)

11 Summer 2006 I2T2 Probability & Statistics Page 175 Simulating an epidemic: NY Standards: 7.S.1, 4, 6; A.S.7, 8; A2.S.6, 7 To simulate this in the classroom, assign a number to each person and have everyone stand up. For demonstration purposes, we shall use n = 10 people. The larger the number, the better the statistics generated. We want to generate random integers from 1 to n, count how many people are infected after each time period (depending on the disease, this could be 1 day, 1 week, 1 month, etc.), and record this number in a table (see below). For our demonstration, we will use randint(1,10). To start press Enter once to get the first random integer. result 8 Person #8 is now infected and should sit down. In row #1 of the table, enter 1. Press Enter once. result 2 Person #2 is now infected and should sit down. In row #2 of the table, enter 2 for number infected. Now we will modify the function to read randint(1,10,2) because each of these infected people could infect two additional people. result 4 1 Persons #4 and #1 are now infected and should sit down. In row #3 of the table, enter 4 for number infected. Press 2nd ENTRY to modify out function to randint(1,10,4) because we have 4 infected people. result These people should now sit down. In row #4 of the table, enter 8 for number infected. Modify the function to read randint(1,10,8) because we have 8 infected people. result Person #6 is now infected and should sit down.. Persons #1, #3, and #2 were already infected. In row #5 of the table, enter 9 for number infected. Our function is now randint(1,10,9) because we have 9 infected people. result Person #9 is now infected and should sit down. All the rest were already infected. In row #6 of the table, enter 10 for number infected. time random # number infected , , 7, 3, With a different size group, you might need more or fewer time periods. Having students sit down makes it easier to count how many are infected. You could also list all the number and cross them off as people are infected. Typically the number newly infected starts slowly, then increases rapidly, and then slows down again as most of the population is infected.

12 Summer 2006 I2T2 Probability & Statistics Page 176 Go to STAT and Edit. Clear L1 and L2. To analyze this data, enter the time in L1 in the calculator. Enter the number infected in L2. Go to 2nd STAT PLOT, turn off all plots except for Plot 1. Turn Plot 1 On. Arrow down to Type and choose the first plot, a scatter plot, press Enter. For Xlist, press 2nd 1 to choose L1, Enter. For Ylist, enter L2. Choose the + mark. Now press Y= to clear out any equations there. Press ZOOM, then ZoomStat to set the window. For our small sample, the data looks sort of linear. You can experiment with various regression equations to see what fits best. We do want the equation pasted into Y1, so on go to VARS, Y-VARS, Function, Y1. The r-value is.98 for our limited data, which indicates a good fit. Press GRAPH to see both the data and the equation. We know that an epidemic is not linear. The appropriate regression equation is the logistic equation. Go to STAT CALC, and to Logistic, Enter. L1 and L2 are the defaults, so go to VARS, Y-VARS, Function, Y2, and Enter. Press Enter to calculate this will take a while. The logistic formula is c "bx, which is not something you would find in your typical textbook. The 1 + a! e curve is S shaped. Go to Y= and turn off Y1 by positioning the cursor over the = sign and pressing Enter. Now press GRAPH. Even with our small sample we can see that the logistic curve is a better model for this type of data.

13 Summer 2006 I2T2 Probability & Statistics Page 177 Final exam scores for 50 students in General Chemistry The Five-Number Summary group of data consists of the following five numbers: 1. L, the smallest value in the data set 2. Q 1, the first quartile (The first quartile is the 25 th percentile.) 3. Q 2, the median 4. Q 3, the third quartile (The third quartile is the 75 th percentile.) 5. H, the largest value in the data set The difference between the first and third quartiles is called the interquartile range. It is the range of the middle 50% of the data. A box-and-whiskers display is a graphic representation of the 5-number summary. The box represents the middle half of the data that lies between the two quartiles. The whiskers are line segments used to represent the other half of the data. One line segment represents the quarter of the data that is smaller than the 1 st quartile. The other line segment represents the quarter of the data that is larger than the 3 rd quartile. A vertical line is placed in the box at the location of the median.

14 Summer 2006 I2T2 Probability & Statistics Page 178 Box and Whisker Plots To create a box and whisker plot for data in List 1, press [2 nd ][Y=] and choose Plot1. Then set the calculator for a boxplot. Press [ZOOM] and choose 9: ZoomStat to let the calculator choose an appropriate viewing window. Press [GRAPH] to see the box and whisker plot. Suppose the boxplot below represents the final exam grades for 50 chemistry students. Write a paragraph discussing how the students did on the exam.

15 Summer 2006 I2T2 Probability & Statistics Page 179 What s My Data? 1. For each of the box and whisker plots, create a 7-value data set that would create it. Warning: You may need to change the automatic window settings.