Download 1, the following data are taken from a certain heat

1. The following data are taken from a certain heat transfer test. The expected
correlation equation is y=axb. Plot the data in an appropriate manner and use the
method of least squares to obtain the best correlation.
X
2040
2580
2980
3220
3870
1690
2130
2420
2900
3310
1020
1240
1360
1710
2070
Y
33.2
32.0
42.7
57.8
126.0
17.4
21.4
27.8
52.1
43.1
18.8
19.2
15.1
12.9
78.5
(a) find a, b and da ( or square root of “a”), db (or square root of “b”)
(b) Calculate the mean deviation of these data from the best correlation
Part a
The best way to approach these problem is to first linearise the correlation equation:
y=axb
Take ln of both sides
lny= lna + blnx
Hence we have a linear regression equation model for this problem. All we do now is plot
the lny vs. lnx in Excel, and find the respective coefficients. I have plotted this in Excel
file attached.
Once you plotted the series, select the Chart, go to Chart Options, Add trendline, select
Linear and then Options, Display equation and R2 on chart.
The equation that we got is:
lnY=1.29lnx – 6.44
The values of a, b, da, db are calculated in the excel file.
a
b
0.0016
da
3.63
db
0.04
1.91
Part b
Alternatively, using Excel's built-in regression analysis package (Tools-Data AnalysisRegression), the following output is generated in Sheet 1.
Regression Statistics
Multiple R
0.772636217
R Square
0.596966724
Adjusted R
Square
0.565964164
Standard Error
0.428748389
Observations
15
We see that the standard error is 0.4287. This is also the mean deviation from the best
correlation because standard error is a measure of the data scatter about the best-fit line,
and has the same units as y itself.
2) for the following data points y is expected to be a power law function of x. obtain this quadratic
function by means of a graphical plot and also by the method of least squares: function: y=a0 X
^(a1)
X
1
2
3
4
5
y
1.9
9.3
21.5
42.0
115.7
a) find a0, a, square root of “a” that gives the best fit for the data.
b) Plot the data
Let call the function y=axb to be easy
Again, linearise the equation
lny= lna + blnx
I have plotted this in the question 2 sheet in Excel.
The equation is
Lny=2.43lnx + 0.56
The values of a, b, da, db are calculated in the excel file.
a
b
1.7507
da
11.36
db
1.32
3.37
The plot is:
Plot of lny vs. lnx
5.00
y = 2.43x + 0.56
R2 = 0.98
4.50
4.00
3.50
3.00
lny 2.50
2.00
1.50
1.00
0.50
0.00
0.00
0.50
1.00
1.50
lnx
Reference:
http://www.mne.psu.edu/me82/Learning/Stat_2/stat_2.html
2.00