This exploration will lend insight to other aspects of using the calculator to stimulate mathematical thinking.

The table below shows the population,P, of India (in millions), andt, the time in years after 1960.

t0 1 2 3 4 5 6 P67.38 69.13 70.93 72.77 74.67 76.61 78.60

We will enter this data into a table on the calculator and look at the scatterplot that it generates.

- Keystroke sequence:

[STAT] >> EDIT >> 1:Edit >> [ENTER]

This takes you to a screen that shows lists as columns, labeled L1, L2, L3.

- If the list has values showing, you can clear them out using this sequence:

[UP arrow until the label is highlighted] >> [CLEAR] >> [ENTER]

- L1 is the
defaultlocation for the independent variable (horizontal, or x- axis); we will use that to show the time,t.

When the list is empty, the cursor begins in the first position. The bottom line on the screen should show "L1(1)=". Key in "0" and press [ENTER]. The "0" will appear as the first item in L1. Put 1 through 6 in as L1(2) through L1(7).

- L2 is the
defaultlocation for the dependent variable (vertical, or y- axis); we will use that to show the population,P.

Enter the population values in, starting with 67.38 as L2(1) and 78.60 as L2(7). If you want to remove or add a number in a list use the [DEL]ete key or [2nd]-[INS]ert key.

- Can you determine an appropriate window to view all of the data points? For convenience, there is a built-in feature that automatically defines and opens an efficient graph window that will show all data points in your lists:

[ZOOM] >> 9:ZoomStat >> [ENTER].

- A
regression equationis an equation that will generate abest fitgraph that represents the data as closely as possible. We can get the calculator to find the best fit for different types of functions, such as: linear (y = ax + b), quadratic (2nd degree), cubic (3rd degree), quartic (4th degree), exponential (y = ab^{x}), power (y = ax^{b}), and others.

Let's find the line of best fit for this data and see how it compares to our data points. To do this, use the sequence:

[STAT] >> [RIGHT arrow] >> 4:LinReg (ax+b) >> [ENTER] >> [ENTER] again.

You should get something that says "a =...", "b =...", "r^{2 }=...", "r =...".

Briefly, the "r value" is a correlation coefficient; it tells how well the regression equation fits the data -- where a "0" is the worst fit, and the closer the absolute value is to "1", the better the fit.

- We'd like to graph the regression equation along with the plotted points. The usual approach is to calculate the regression equation, then paste it into a Y= slot. To do this use the sequence:

[STAT] >> CALC >> 4:LinReg(ax+b) >> [ENTER] >> [ENTER], which leaves the screen showing values for a, b, r^{2}, and r. Go to [Y=], clear out Y_{1}and leave the cursor there as a "beacon". Now use [VARS] >> 5:Statistics >> [ENTER] >> EQ >> 1:RegEQ >> [ENTER] to paste the equation into Y_{1}.

How well does the line match with the data points? What is the meaning of the slope and intercept here?

- The slope of a linear function is really just a constant rate of change. If you calculate the slope between adjacent points, do you get the same values?

- Here is additional information about India's population: in 1970 the population was 87.10 million, which increased to 112.58 by 1980, reached 145.53 million in 1990, and by 2000 the population had grown to 188.12 million. How does this affect the apparent fit of the linear regression?

Enter these new points by going to [STAT] >> EDIT >> 1:Edit; now scroll down and add the new values to L1 and L2. Press Graph. Why might the new data points not show in our graph? What can we do to get them to show?

- The linear regression that seemed to fit the data earlier now seems considerably off. Perhaps this population is
notgrowing at a constant rate. In fact, many naturally occuring growth and decay situations are best modeled by exponential functions of the form:

Population_{(at time t)}= {Initial population_{(t = 0)}} x {Growth Rate}^{t}

Let's try finding and graphing the exponential regression equation to see how well that models the data. This time we can use an alternate sequence of keystrokes:

[STAT] >> CALC >> (8 or 0):ExpReg >> [2nd]-[L1] >> [,] >> [2nd]-[L2] >> [,] >> [VARS] >> Y-VARS >> 1:Function >> [ENTER] >> [specify Y_{2}] >> [ENTER] >> [ENTER].

As before, this takes you to a screen that shows the values for a, b, r^{2}, and r, but the equation has automatically been pasted into the specified Y_{2}slot. To see the graph, press [GRAPH]. First the data points appear, then the "Y_{n}=" functions will appear in their numerical order.

- What can you conclude from this exploration?

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