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
Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from
Qd 120 3P
to
Qd 90 3P
and the supply function has shifted from
QS
20 2P
to
QS
10 2P .
a) Find the effect of this change on price and quantity.
b) Which of the changes in demand and supply is higher?
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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