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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

where do I start
Jeremy Reply
in a large restaurant an average of every 7 customers ask for water with the their meal. A random sample of 12 customers is selected, find the probability that exactly 6 ask for water with their meal
Rufai Reply
any body with idea
Rufai
conditional probability
Ramesh
can you help please
Rufai
iam really sorry. it's been long since I used these things. I just gave you a hint though
Ramesh
ok
Rufai
this follows binomial distribution. p(X=6)=12C6*(0.6)^6*0.4^6 use this formula n find.
syeda
can you explain the cosidered variable in the formula
Divya
x is variable wich is exactly 6 costumers
syeda
n is number of customers
syeda
ncx*p^X*q^X?
Divya
q^n-x
syeda
oh right !!! thanks yaar
Divya
I agree with Seyda too
Hoshyar
I agree with Syeda too
Hoshyar
7/12 =0.58is it?
yousaf
.
yousaf
r8
khalid
what is descriptive statistic
Shadow Reply
Descriptive statistics are brief descriptive coefficients that summarize a given data set, which can be either a representation of the entire or a sample of a population. Descriptive statistics are broken down into measures of central tendency and measures of variability (spread). 
Divya
are you getting this ?
Divya
if so let me know
Divya
yes m getting
Ramesh
fine
Divya
what's taking place can l join u
Anest
yeah !!why not? sure
Divya
okey thanks
Anest
where are statistics used
akinola Reply
hello
Giannis
Hi
Makhosi
how u doing
Muhid
please guys uses of statistics....
james Reply
the upper quartile of the population 10,12,14,16,18,20,25,15,11,11,17,is................?
Gach
The probability range is 0 to 1... but why we take it 0 to 1....
Muhammad Reply
because in probability 1 means success and 0 means failure and it cnnt be more or less than 1 and 0.
syeda
b/c v hv mazimum probibliy 1 and minimum which is.no.probiblity is 0.so.v hv the range from 0 to 1
khalid
the size of a set is greeter than its subset
Hoshyar
The probability of an event will not be less than 0.  This is because 0 is impossible (sure that something will not happen).The probability of an event will not be more than 1. This is because 1 is certain that something will happen
Divya
what do they mean in a question when you are asked to find P40 and P88
Megrina Reply
I dont get your question! What are you talk ING about?
Mani
hi
Mehri
you're asked to find page 40 and page 88 on that particular book.
Joseph
hi
ravi
any suggestions for statistics app better than this
ravi
sorry miss wrote the question
omar
No problem) By the way. I NEED a program For statistical data analysis. Any suggestion?
Mani
Eviews will help u
Kwadwo
Hello
Okonkwo
arey there any data analyst and working on sas statistical model building
ravi
Hi guys ,actually I have dicovered that the P40 and P88 means finding the 40th and 88th percentiles 😌..
Megrina
who can explain the euclidian distance
ravi
I am fresh student of statistics (BS) plz guide me best app or best website relative to stat topics
Noman
IMAGESNEWSVIDEOS A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are the mean, median, and mode.
Ahmed Reply
define the measures of location
Kaynaat Reply
IMAGESNEWSVIDEOS A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are th
Ahmed
hi i have a question....
Muhammad
what is confidence interval estimate and its formula in getting it
Jhezarie Reply
discuss the roles of vital and health statistic in the planning of health service of the community
BITRUS Reply
given that the probability of
BITRUS
good morning
Rufai
hi bitrus.
Bratit
hi
yousaf
hallo
Bratit
hey
Divya
uses in health policy and health research
Divya
1) social inequalities I health 2) Trends in mortality
Divya
hey do you know any about any of the statistics courses which is best ?
Divya
hello
Hoshyar
hi
Hoshyar
online courses and convenient fees
Divya
can man city win Liverpool ?
Emmanuel Reply
There are two coins on a table. When both are flipped, one coin land on heads eith probability 0.5 while the other lands on head with probability 0.6. A coin is randomly selected from the table and flipped. (a) what is probability it lands on heads? (b) given that it lands on tail, what is the Condi
Nusrat Reply
0.5*0.5+0.5*0.6
Ravasz
what is gradient descent?
Saurav Reply
It should be a Machine learning terms。
Mok
it is a term used in linear regression
Saurav
what are the differences between standard deviation and variancs?
Enhance
what is statistics
Emmanuel Reply
statistics is the collection and interpretation of data
Enhance
the science of summarization and description of numerical facts
Enhance
Is the estimation of probability
Zaini
mr. zaini..can u tell me more clearly how to calculated pair t test
Haai
do you have MG Akarwal Statistics' book Zaini?
Enhance
Haai how r u?
Enhance
maybe .... mathematics is the science of simplification and statistics is the interpretation of such values and its implications.
Miguel
can we discuss about pair test
Haai
hello can i join
Kawuma
Hi. Im raph. Can anyone teach me how to read the z-table?
Asare

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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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