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  • Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. 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 zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

Drawing a conclusion:

There are two methods of making the decision. The two 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%, α = 0.05

Note

Using the p -value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 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

To calculate the p -value using LinRegTTEST:
On the LinRegTTEST input screen, on the line prompt for β or ρ , 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 zero."

    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 zero."

    Calculation notes:

  • You will use technology to calculate the p -value. The following describes the calculations to compute the test statistics and the p -value:
  • The p -value is calculated using a t -distribution with n - 2 degrees of freedom.
  • The formula for the test statistic is t = r n 2 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: ŷ = -173.51 + 4.83 x 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 0 : 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 0
  • Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Questions & Answers

what is ch-square test
Muluken Reply
hindi notes please😫🙏🙏💓
Uttam Reply
hindi notes please😫🙏🙏💓
Uttam
How can I calculate the Class Mark, Relative frequency and the cumulative frequency on a frequency table?
IJOGI Reply
what is the important in business planning and economics
mahelt Reply
explain the limitation and scope of statistics
mahelt
statistics is limited to use where data can be measured quantitatively. statistics scope is wider such as in economic planning, medical science etc.
Gurpreet
can you send me mcq type questions
Tanuj Reply
Yas
Umar
which books are best to learn applied statistics for data science/ML
Gurpreet
A population consists of five numbers 2,3,6,8,11.consists all possible samples of size two which can be drawn with replacement from this population. calculate the S.E of sample means
Karunesh Reply
A particular train reaches the destination in time in 75 per cent of the times.A person travels 5 times in that train.Find probability that he will reach the destination in time, for all the 5 times.
Anish Reply
0.237
Amresh
explain that this answer-0.237
umesh
p(x=5)= 5C0 p^5 q^0 solve this
Amresh
please sir.
umesh
ok
umesh
5C0=1 p^5= (3/4)^5 q^0=(1/4)^0
Amresh
Hint(0.75 in time and 0.25 not in time)
kamugi
what is standard deviation?
Jawed Reply
It is the measure of the variation of certain values from the Mean (Center) of a frequency distribution of sample values for a particular Variable.
Dominic
what is the number of x
Godgift Reply
10
Elicia
Javed Arif
Jawed
how will you know if a group of data set is a sample or population
Kingsley Reply
population is the whole set and the sample is the subset of population.
umair
if the data set is drawn out of a larger set it is a sample and if it is itself the whole complete set it can be treated as population.
Bhavika
hello everyone if I have the data set which contains measurements of each part during 10 years, may I say that it's the population or it's still a sample because it doesn't contain my measurements in the future? thanks
Alexander
Pls I hv a problem on t test is there anyone who can help?
Peggy
What's your problem Peggy Abang
Dominic
Bhavika is right
Dominic
what is the problem peggy?
Bhavika
hi
Sandeep
Hello
adeagbo
hi
Bhavika
hii Bhavika
Dar
Hi eny population has a special definition. if that data set had all of characteristics of definition, that is population. otherwise that is a sample
Hoshyar
three coins are tossed. find the probability of no head
Kanwal Reply
three coins are tossed consecutively or what ?
umair
p(getting no head)=1/8
umair
or .125 is the probability of getting no head when 3 coins are tossed
umair
🤣🤣🤣
Simone
what is two tailed test
Umar Reply
if the diameter will be greater than 3 cm then the bullet will not fit in the barrel of the gun so you are bothered for both the sides.
umair
in this test you are worried on both the ends
umair
lets say you are designing a bullet for thw gun od diameter equals 3cm.if the diameter of the bullet is less than 3 cm then you wont be able to shoot it
umair
In order to apply weddles rule for numerical integration what is minimum number of ordinates
Anjali Reply
excuse me?
Gabriel
why?
Tade
didn't understand the question though.
Gabriel
which question? ?
Tade
We have rules of numerical integration like Trapezoidal rule, Simpson's 1/3 and 3/8 rules, Boole's rule and Weddle rule for n =1,2,3,4 and 6 but for n=5?
John
Someone should help me please, how can I calculate the Class Mark, Relative frequency and the cumulative frequency on a frequency table?
IJOGI
geometric mean of two numbers 4 and 16 is:
iphone Reply
10
umair
really
iphone
quartile deviation of 8 8 8 is:
iphone
sorry 8 is the geometric mean of 4,16
umair
quartile deviation of 8 8 8 is
iphone
can you please expalin the whole question ?
umair
mcq
iphone
h
iphone
can you please post the picture of that ?
umair
how
iphone
hello
John
10 now
John
how to find out the value
srijth Reply
can you be more specific ?
umair
yes
KrishnaReddy

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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