1. Visualizing the data. Using graph paper, complete the following steps:

a) Use the horizontal axis (x) to represent the year. Begin with year 1960 and continue through year 2050. You will need to choose an appropriate scale for the x-axis to model the data.

b) Use the vertical axis (y) to represent the times, in seconds. You will first need to convert all the times to seconds instead of minutes and seconds. Then choose an appropriate scale for the y-axis to model the data, beginwith 200 seconds and continue to 300 seconds.

c) Use a plus (+) sign to represent the data for men, and plot the points on the grid using the year as the x coordinate and the times for men as the y coordinate. Use an "o" to represent the data for women, and plot the points on the same grid using the year as the x coordinate and the times for women as the y coordinate.

d) Title your graph and label the x and y axes.

2. Finding the models

a) Using the years as x and the times in seconds for men as y, enter the data in

your calculator in L₁ and L₂, respectively. Do the same for women except use L₃ and L₄ .

b) Set up the plots in your calculator so that you can display L₁ and L₂ as Plot1. Set

up Plot2 to display L₃ and L₄. [Recall that the setup for plots is done with 2^nd Y=. Make sure that each plot uses a different mark so that you can distinguish between the two.] Use Zoom 9 to view the scatter plot, once you have set up the plots. Then adjust the window to match the grid on your graph paper. It should resemble the one you drew in part I.

c) Using the Calculate menu under Stats, find a linear regression model for L₁ and L₂. Write the corresponding equation, showing accuracy to threedecimal places for a and b.

d) Using the Calculate menu under Stats, find a linear regression model for L₃ ??and L₄. Note that the format will be LinReg (y=ax+b) (L₃,L₄). Write the ???corresponding equation, showing accuracy to three decimal places for a and b.

e) Draw the two lines whose equations you found in parts c and d on the graph ?you

produced in part I. A simple way to do this is to find two points on each line

by substituting in values for x and solving for y. Use years that are actually on the

grid, i.e. between 1960 and 2050. Do the lines appear to intersect? Where? Show approximate coordinates, based on the grid, as integers.

III. Interpreting the models

a) Using the above models, find the year for which the time for men and the time for women are the same. (Toanswer this question, solve the system of equations using both model equations. Show all work! Round the year to the nearest whole number and time to 3 decimal places.) Compare your answer to the one you found in part IIeabove. How close is your answer from the solving the system of equations to your approximate answer from looking at the graphs? Convert the time from seconds to minutes and seconds. Do you think the time is reasonable? Why or why not?

b) Use your models to predict the world record times for men and women in 2020, 2050, and 2270. Convert the times from seconds to minutes and seconds. Are the predictions reasonable? Why or why not?

d) Interpret the slopes of both regression lines. (Use the method we discussed in class.) Based on the slopes, whose times are improving more as the years go by, men or women?

e) Find and interpret the y-intercepts of the models. Do the y-intercepts give any meaningful information? Why or why not?

f) The data shows that men run the mile faster than women. According to the models, will that always be the case? Do the models predict that women will at some point run the mile faster than men? If so, do you think that is a reasonable prediction? Why or why not?

g) Based on the above and on any other pertinent mathematical or non-mathematical considerations, discuss the validity of the models and their use as predicting tools. Consider the following: Over what years would it make sense to use the models to predict world record times for men and women? Is there another shape of a graph that might be a better model for the data? What shape would you suggest?

?

I

The following are 13 world record times for men and women in the mile race as ratified by the International Association of Athletics Federations. The mile is the only non-metric distance that is recognized for record purposes. The times are given in minutes and seconds. Notice times prior to 1981 are accurate to a tenth of a second. Beginning in 1981, IAAF began recognizing times to a hundredth of a second. Also notice that the years of the records are not the same for men and women. Data from Wikipedia.org/wiki/mile_run_world_record_progression Year Men's Times Athlete's Nationality 1965 3:53.6 France 1966 3:51.3 US 1967 3:51.1 US 1975 3:51.0 Tanzania 1975 3:49.4 New Zealand 1979 3:49.0 UK 1980 3:48.8 UK 1981 3:48.53 UK 981 3:48.40 UK 1981 3:47.33 UK 1985 3:46.32 UK 1993 3:44.39 Algeria 1999 3:43.13 Morocco Year Women's Times Athlete's Nationality 1967 4:37.0 UK 1969 4:36.8 Netherlands 1971 4:35.3 West Germany 1973 4:29.5 Italy 1977 4:23.8 Romania 1979 4:22.09 Romania 1980 4:21.68 US 1981 4:20.89 Soviet Union 1982 4:18.08 US 1982 4:17.44 Romania 1985 4:16.71 US 1989 4:15.61 Romania 1996 4:12.56 Russia