Determine the linear correlation coefficient. The linear correlation coefficient is r = 0.99. (Round to three decimal places as needed.) Determine the null and alternative hypotheses.
Answer (1)
To determine the linear correlation coefficient and the hypotheses, let's go through the steps.
The linear correlation coefficient, often represented by the letter r , measures the strength and direction of a linear relationship between two variables on a scatter plot. Its value ranges from -1 to 1.
For the given value r = 0.990 , this indicates a very strong positive linear relationship between the two variables being analyzed.
Next, we'll define the null and alternative hypotheses. These are typically used in hypothesis testing to make inferences about the population parameter, ρ (the population correlation coefficient).
Null Hypothesis ( H 0 ) : This hypothesis usually states that there is no effect or no relationship. In the context of correlation, it is often that the population correlation coefficient is zero.
H 0 : ρ = 0
This implies there is no linear relationship between the two variables in the population.
Alternative Hypothesis ( H a ) : This hypothesis is what you want to prove. It says that there is a relationship. Depending on the research question, this could be that there is a positive correlation, a negative correlation, or simply a correlation (not zero).
If you wish to test if the correlation is different from zero (i.e., there is a correlation but not specifying the direction):
H a : ρ = 0
If you specifically want to test for positive correlation:
0"> H a : ρ > 0
If you want to test for a negative correlation:
H a : ρ < 0
In this scenario, since the linear correlation coefficient r is 0.990, which is very close to 1, the typical alternative hypothesis you might consider is 0"> ρ > 0 , to match the strong positive correlation observed.
In summary, given that r = 0.990 , the null hypothesis would generally suggest no correlation, while a reasonable alternative hypothesis could assert a positive correlation due to the observed data.
