For each data point, you can calculate the residuals or errors,
${y}_{i}-{\hat{y}}_{i}={\epsilon}_{i}$ for
$i=\text{1, 2, 3, ..., 11}$ .
Each
$\left|\epsilon \right|$ is a vertical distance.
For the example about the third exam scores and the final exam scores for the 11
statistics students, there are 11 data points. Therefore, there are 11
$\epsilon $ values. If you
square each
$\epsilon $ and add, you get
Using calculus, you can determine the values of
$a$ and
$b$ that make the
SSE a minimum. When you make the
SSE a
minimum, you have determined the points that are on the line of best fit. It turns out thatthe line of best fit has the equation:
$\hat{y}=a+\text{bx}$
where
$a=\overline{y}-b\cdot \overline{x}$
$\overline{x}$ and
$\overline{y}$ are the sample means of the
$x$ values and the
$y$ values, respectively. The best fit line always passes through the point
$(\overline{x},\overline{y})$ .
and
$b=r\cdot \left(\frac{{s}_{y}}{{s}_{x}}\right)$
where
${s}_{y}$ = the standard deviation of the
$y$ values and
${s}_{x}$ = the standard deviation of the
$x$ values.
$r$ is the correlation
coefficient which is discussed in the next section.
Least squares criteria for best fit
The process of fitting the best fit line is called
linear regression . The idea behind finding the best fit line is based on the assumption that the data are
scattered about a straight line. The criteria for the best fit line is that the sum of the squared errors (SSE) is minimized, that is made as small as possible. Any other line you might choose would have a higher SSE than the best fit line. This best fit line is called the
least squares regression line .
Using the summary statistics and correlation coefficient for the relationship between third exam score and final exam score we will calculate the regression equation, the line of best fit.
The least squares regression line (best fit lint) for the third exam/final exam example has the equations:
$\hat{y}=-\mathrm{173.49}+4.83x$
Third exam vs final exam example:
The graph of the line of best fit for the third exam/final exam example is shown below:
Remember, it is always important to plot a
scatter diagram first. If the scatter plot indicates that there is a linear relationship betweenthe variables, then it is reasonable to use a best fit line to make predictions for
$y$ given
$x$ within the domain of
$x$ -values in the sample data,
but not necessarily
for
$x$ -values outside that domain.
You could use the line to predict the final exam score for a student who earned a grade of 73 on the third exam.
You should NOT use the line to predict the final exam score for a student who earned a grade of 50 on the third exam, because 50 is not within the domain of the x-values in the sample data, which are between 65 and 75.
Understanding slope
The slope of the line, b, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.
INTERPRETATION OF THE SLOPE: The slope of the best fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.
Third exam vs final exam example
Slope: The slope of the line is b = 4.83.
Interpretation: For a one point increase in the score on the third exam, the final exam score increases by 4.83 points, on average.
Questions & Answers
Do somebody tell me a best nano engineering book for beginners?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Source:
OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
Google Play and the Google Play logo are trademarks of Google Inc.
Notification Switch
Would you like to follow the 'Collaborative statistics using spreadsheets' conversation and receive update notifications?