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The correlation coefficient, r , tells us about the strength and direction of the linear relationship between x and y . However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n , together.

We perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute r , the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r , is our estimate of the unknown population correlation coefficient.

  • The symbol for the population correlation coefficient is ρ , the Greek letter "rho."
  • ρ = population correlation coefficient (unknown)
  • r = sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is "close to zero" or "significantly different from zero". We decide this based on the sample correlation coefficient r and the sample size n .

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is "significant."

  • 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.
  • What the conclusion means: There is a significant linear relationship between x and y . We can use the regression line to model the linear relationship between x and y in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is "not significant".

  • 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."
  • What the conclusion means: There is not a significant linear relationship between x and y . Therefore, we CANNOT use the regression line to model a linear relationship between x and y in the population.

Note

  • If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
  • If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
  • If r is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed x values in the data.

Performing the hypothesis test

  • Null Hypothesis: H 0 : ρ = 0
  • Alternate Hypothesis: H a : ρ ≠ 0

What the hypotheses mean in words:

Questions & Answers

how do you find z if you only know the area of .0808
Cady Reply
How to take a random sample of 30 observations
Hamna Reply
you can use the random function to generate 30 numbers or observation
smita
How we can calculate chi-square if observed x٫y٫z/frequency 40,30,20 Total/90
Insha Reply
calculate chi-square if observed x,y,z frequency 40,30,20total 90
Insha
find t value,if boysN1, ،32,M1,87.43 S1square,39.40.GirlsN2,34,M2,82.58S2square,40.80 Determine whether the results are significant or insignificant
Insha
The heights of a random sample of 100 entering HRM Freshman of a certain college is 157 cm with a standard deviation of 8cm. test the data against the claim that the overall height of all entering HRM students is 160 cm. previous studies showed that
Crispen Reply
complete the question.. as data given N = 100,mean= 157 cm, std dev = 8 cm..
smita
Z=x-mu/ std dev
smita
find the mean of 25,26,23,25,45,45,58,58,50,25
Asmat Reply
add all n divide by 10 i.e 38
smita
38
hhaa
amit
1 . The “average increase” for all NASDAQ stocks is the:
Jamshaid Reply
STATISTICS IN PRACTICE: This is a group assignment that seeks to reveal students understanding of statistics in general and it’s practical usefulness. The following are the guidelines; 1.      Each group has to identify a natural process or activity and gather data about/from the process. 2.     
Kofi Reply
The diameter of an electric cable,say, X is assumed to be continoues random variable with p.d.f f(x)=6x(1-x); ≤x≤1 a)check that f(X) is p.d.f b) determine a number b such that p(Xb)
Syed Reply
A manufacturer estimate 3% of his output is defective. Find the probability that in a sample of 10 items (a) less than two will be defective (b) more than two will be defective.
ISAIAH Reply
A manufacturer estimates that 3% of his output of a small item is defective. Find the probabilities that in a sample of 10 items (a) less than two and (b) more than two items will be defective.
ISAIAH
use binomial distribution with parameter n=10, p= 0.03, q=0.97
Shivprasad
the standard deviation of a symmetrical distribution is 7.8 . what must be the value of forth moment about the mean in order that distribution be a) leptokurtic b) mesokurtic c) platy kyrtic intrept the obtain value of a b and c
Tushar Reply
A researcher observed that four out of every ten of their products are normally defective. A total of 360 samples of the products were being tested. If the sample is normally distributed and 220 of the products were identified to be faulty, test the hypothesis that the observation of the res
Adepoju Reply
false
HariPrasad
please answer the ques"following values are obtained from life table T15=3,493,601 and e°15=44.6 then expected number of person alive at exact age 15 will be "
vinay
make it clear
Kagimu
how x minus x bar is equal to zero
Kashif Reply
When the mean (X bar) of the sample and the datapoint-in-context (X) from the same sample are the same, then it (X minus X bar) is equal to 0
Johns
e.g. mean of. sample is 3 and one of the datapoints in that sample is also 3
Johns
a numerical value used as a summary measure for a sample such as a sample mean is known as
rana Reply
differentiate between qualitative and quantitative variables
rana Reply
qualitative variables are descriptive while quantitative are numeric variables
Chisomo
please guys what is the formulas use in calculated statistics please iam new here
Yunisa Reply
Dear Yunisa there are different formulas used in statistics depending on wnat you want to measure. It would be helpful if you can be more specific
LAMIN

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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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