13.6 Facts about the correlation coefficient for linear regression  (Page 2/3)

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Setting up the hypotheses:

• Null Hypothesis: ${H}_{o}$ : $\rho$ = 0
• Alternate Hypothesis: ${H}_{a}$ : $\rho$ ≠ 0

What the hypotheses mean in words:

• Null Hypothesis ${H}_{o}$ : The population correlation coefficient IS NOT significantly different from 0. There IS NOT a significant linear relationship(correlation) between $x$ and $y$ in the population.
• Alternate Hypothesis ${H}_{a}$ : The population correlation coefficient IS significantly DIFFERENT FROM 0. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between $x$ and $y$ in the population.

Drawing a conclusion:

• There are two methods to make the decision. Both methods are equivalent and give the same result.
• Method 1: Using the p-value
• Method 2: Using a table of critical values
• In this chapter of this textbook, we will always use a significance level of 5%, $\alpha$ = 0.05
• Note: Using the p-value method, you could choose any appropriate significance level you want; you are not limited to using $\alpha$ = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, $\alpha$ = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

Method 1: using a p-value to make a decision

• The linear regression $t$ -test LinRegTTEST on the TI-83+ or TI-84+ calculators calculates the p-value.
• On the LinRegTTEST input screen, on the line prompt for $\beta$ or $\rho$ , highlight " ≠ 0 "
• The output screen shows the p-value on the line that reads "p =".
• (Most computer statistical software can calculate the p-value.)

If the p-value is less than the significance level (α = 0.05):

• Decision: REJECT the null hypothesis.
• 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."

If the p-value is not less than the significance level (α = 0.05)

• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is NOT significantly different from 0."

Calculation notes:

• You will use technology to calculate the p-value. The following describe the calculations to compute the test statistics and the p-value:
• The p-value is calculated using a $t$ -distribution with $\mathrm{n-2}$ degrees of freedom.
• The formula for the test statistic is $t=\frac{r\sqrt{n-2}}{\sqrt{1-r^{2}}}$ . The value of the test statistic, $t$ , is shown in the computer or calculator output along with the p-value. The test statistic $t$ has the same sign as the correlation coefficient $r$ .
• The p-value is the combined area in both tails.
• An alternative way to calculate the p-value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

Third exam vs final exam example: p value method

• Consider the third exam/final exam example .
• The line of best fit is: $\stackrel{^}{y}=-173.51+\text{4.83x}$ with $r=0.6631$ and there are $\mathrm{n = 11}$ 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)?

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
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Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
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
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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
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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
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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