### Introduction

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 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 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 the correlation coefficient is significantly different from zero, we say the correlation coefficient is *significant*.

- Conclusion: There is sufficient evidence to conclude 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 the correlation coefficient is not significantly different from zero (it is close to zero), we say the correlation coefficient is *not significant*.

- Conclusion: There is insufficient evidence to conclude 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 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.