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The table shows the depth (d metres) of water in a harbour at certain times (t hours) after midnight on a particular day.

time t (hours)
0
1
2
3
4
5
6
7
8
9
10
11
12
13

and the in the next column corresponding to the figures above is
depth d (m)
3.0
3.3
4.2
5.6
7.2
8.2
9.0
8.9
8.1
7.3
5.6
4.3
3.5
3.1

Use the regression facilities on your calculator to fit a sine curve to these data. Choose the one option which provides the best fit model (with coefficient rounded to 2 significant figures).
A) t=3.0sin(0.49d-1.6)+6.0
B) d=3.0+6.0sin(0.49t-1.6)
C) t=2.9sin(0.48d-1.5)+6.0
D) d=3.0sin(0.49t-1.6)+6.0
E) d=2.99sin(0.49t-1.60)+6.04
F) d=6.0-3.0sin(0.49t+1.6)

Can anyone help me please. I may be d?

Note that you want d(t) to be of the form:

d(t) = A + B sin(r t + s)

Now, as you can see, all the options that express d like this have r = 0.49. This makes it easy to find the best fit, because you can now transform this problem into a linear regression problem as follows. You write:


d(t) = A + B sin(r t + s) =

A + B [cos(s) sin(r t) cos(s) + sin(s)cos(rt)(rt)] =

A + B cos(s) sin(rt) + B sin(s) cos(rt)

So, this problem is of the form:

d(t) = A_1 + A_2 f_2(t) +
A_3f_3(t)

with


f_2(t) = sin(rt)

f_3(t) = cos(rt)


So this is a standard linear regression problem as d is a linear function of f_2 and f_3.

You just need to proceed in the same way as you would proceed in case of a regression problem like:

Y = A + B X1 + C X2

for given data for Y, X1, and X2.

i'm not sure what you mean. which are the correct answers?

Count, please email me at bobpursley@gmail.com

Hi Grant,

Do you have a calculator in which you can enter data to do linear regression in multiple variables?

If so, then you need to do the following. Compile a table with the values of X1 = sin(0.49 t) and
X2 = cos(0.49 t).

Then you must enter the data for Y = d, X1 and X2 in your calculator and out will come the regression parameters corresponding to a fit of the form:

Y = A + B X1 + C X2

From B and C you then calculate the amplitude and the phase of the sin...

I've just emailed you :)

I don't have this type of calculator as yet and my homework is due on monday. Is the answer A?

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