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  • H o : ρ = 0
  • H a : ρ ≠ 0
  • α = 0.05
  • The p-value is 0.026 (from LinRegTTest on your calculator or from computer software)
  • The p-value, 0.026, is less than the significance level of α = 0.05
  • Decision: Reject the Null Hypothesis H o
  • Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from 0.
  • Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

Method 2: using a table of critical values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table at the end of this chapter (before the Summary ) may be used to give you a good idea of whether the computed value of r is significant or not . Compare r to the appropriate critical value in the table. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may want to use the line for prediction.

Suppose you computed r = 0.801 using n = 10 data points. df = n - 2 = 10 - 2 = 8 . The critical values associated with df = 8 are -0.632 and + 0.632. If r negative critical value or r > positive critical value , then r is significant. Since r = 0.801 and 0.801 > 0.632 , r is significant and the line may be used for prediction. If you view this example on a number line, it will help you.

Horizontal number line with values of -1, -0.632, 0, 0.632, 0.801, and 1. A dashed line above values -0.632, 0, and 0.632 indicates not significant values.
r is not significant between -0.632 and +0.632. r = 0.801 > +0.632 . Therefore, r is significant.

Suppose you computed r = -0.624 with 14 data points. df = 14 - 2 = 12 . The critical values are -0.532 and 0.532. Since -0.624 -0.532 , r is significant and the line may be used for prediction

Horizontal number line with values of -0.624, -0.532, and 0.532.
r = -0.624 -0.532 . Therefore, r is significant.

Suppose you computed r = 0.776 and n = 6 . df = 6 - 2 = 4 . The critical values are -0.811 and 0.811. Since -0.811 0.776 0.811 , r is not significant and the line should not be used for prediction.

Horizontal number line with values -0.924, -0.532, and 0.532.
-0.811 r = 0.776 0.811 . Therefore, r is not significant.

    Third exam vs final exam example: critical value method

  • Consider the third exam/final exam example .
  • The line of best fit is: y ^ = -173.51 + 4.83x with r = 0.6631 and there are n = 11 data points.
  • Can the regression line be used for prediction? Given a third exam score ( x value), can we use the line to predict the final exam score (predicted y value)?
  • H o : ρ = 0
  • H a : ρ ≠ 0
  • α = 0.05
  • Use the "95% Critical Value" table for r with df = n - 2 = 11 - 2 = 9
  • The critical values are -0.602 and +0.602
  • Since 0.6631 > 0.602 , r is significant.
  • Decision: Reject H o :
  • Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from 0.
  • Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

Additional practice examples using critical values

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if r is significant and the line of best fit associated with each r can be used to predict a y value. If it helps, draw a number line.

  1. r = -0.567 and the sample size, n , is 19. The df = n - 2 = 17 . The critical value is -0.456. -0.567 -0.456 so r is significant.
  2. r = 0.708 and the sample size, n , is 9. The df = n - 2 = 7 . The critical value is 0.666. 0.708 > 0.666 so r is significant.
  3. r = 0.134 and the sample size, n , is 14. The df = 14 - 2 = 12 . The critical value is 0.532. 0.134 is between -0.532 and 0.532 so r is not significant.
  4. r = 0 and the sample size, n , is 5. No matter what the dfs are, r = 0 is between the two critical values so r is not significant.

Assumptions in testing the significance of the correlation coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between x and y in the population.

The regression line equation that we calculate from the sample data gives the best fit line for our particular sample. We want to use this best fit line for the sample as an estimate of the best fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

    The assumptions underlying the test of significance are:

  • There is a linear relationship in the population that models the average value of y for varying values of x . In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
  • The y values for any particular x value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) above implies that these normal distributions are centered on the line: the means of these normal distributions of y values lie on the line.
  • The standard deviations of the population y values about the line are equal for each value of x . In other words, each of these normal distributions of y values has the same shape and spread about the line.
  • The residual errors are mutually independent (no pattern).

A downward sloping regression line is shown with the y values normally distributed about the line with equal standard deviations for each x value. For each x value, the mean of the y values lies on the regression line. More y values lie near the line than are scattered further away from the line.
The y values for each x value are normally distributed about the line with the same standard deviation. For each x value, the mean of the y values lies on the regression line. More y values lie near the line than are scattered further away from the line.

**With contributions from Roberta Bloom

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Principles of business statistics. OpenStax CNX. Aug 05, 2009 Download for free at http://cnx.org/content/col10874/1.5
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