Every one of us is all too familiar with the phenomenon of forgetting.

Question

Facts that we clearly understood at the time we first learned them sometimes fade from our memory by the time the final exam rolls around.

Psychologists have proposed several ways to model this process.

One such model is Ebbinghaus' Law of Forgetting

Other models use exponential or logarithmic functions. To develop her own model, a psychologist performs an experiment on a group of volunteers by asking them to memorize a list of 100 related words. She then tests how many of these words they can recall after various periods of time. The average results for the group are shown in the table.

Time15 min	64.9
1 h	45.5
8 h	37.0
1 day	33.1
2 days	26.3
3 days	25.3
5 days	23.6

right hand column is the words recalled, right hand column is the time

(a) Use a graphing calculator to find a power function of the form y = at^b

 that models the average number of words y that the volunteers remember after t hours. (Round all numerical values to three decimal places.)

Find an exponential function of the form y = ab^t

 to model the data. (Round all numerical values to three decimal places.)

thank you so much!

Answer 1

I'll assume you have a TI-84 calculator.

Hit the "stat" button, then "EDIT", then "1: Edit" and you get a table to enter the data.

In the L1 column, enter your time values but first, convert them all to hours like this:

0.25

1

8

24

48

72

120

Then in the L2 column, enter the "words recalled" data

64.9

45.5

37

33.1

26.3

25.3

23.6

Then push the "stat" button again and scroll right to "CALC" and then down to "A: PwrReg"

Make sure the Xlist says L1 and the Ylist says L2. If it doesn't, select the "list" button to select the proper lists for each. Then scroll down to "Calculate" and hit "enter"

You should get this:

y=a*x^b

a=49.91831543

b=-0.1553015414

(depending on how your settings are set, you might get more info but this is enough to answer your question)

So the equation becomes (letting "words recalled = "W"):

W = 49.918(t)^-0.155 or

W(t) = 49.918(t ^-0.155)

Now repeat the calculation using the "stat" button again, scroll right to "CALC" and then down to "0: ExpReg"

Again, make sure the Xlist says L1 and the Ylist says L2. Then scroll down to "Calculate" and hit "enter"

You should get this:

y=a*b^x

a=44.73322394

b=0.9932801938

So the equation becomes (letting "words recalled = "W"):

W = 44.733(0.993)^t or

W(t) = 44.733(0.933)^t

This turns out to be a crummy curve fit to this data but it's what they wanted so . . .

Answer 2

Using the graphing calculator, the equation is y = ax² +bx + c

a = 0.004

b = -0.722

c = 51.211