To determine the correlation coefficient between average temperature and coat sales, we need to consider what a correlation coefficient represents and how these two variables are likely to be related.
The correlation coefficient is a statistic that measures the direction and strength of the linear relationship between two variables, and it takes on values between -1 and 1:
- A coefficient of 1 implies a perfect positive linear relationship, meaning that as one variable increases, the other variable also increases, and this linear relationship is exact.
- A coefficient of -1 implies a perfect negative linear relationship, meaning that as one variable increases, the other variable decreases, and this linear relationship is exact.
- A coefficient of 0 implies no linear relationship between the variables, meaning changes in one variable do not predict changes in the other.
- Values between 0 and 1 signify varying degrees of positive linear relationship, with values closer to 1 representing stronger relationships.
- Values between 0 and -1 signify varying degrees of negative linear relationship, with values closer to -1 representing stronger relationships.
In the context of average temperature and coat sales, we would reasonably expect that as the average temperature rises, people are less likely to purchase coats since they are primarily used in cold weather. Conversely, as the average temperature falls, coat sales are likely to go up. This forms a classic example of an inverse relationship where one variable tends to decrease as the other increases.
Therefore, we would expect a negative correlation coefficient between average temperature and coat sales because higher temperatures are likely to be associated with lower coat sales. The correlation coefficient in this case would need to be between 0 (no relationship) and -1 (perfect negative linear relationship). Given that no real-world data are perfectly linearly related, we can almost always exclude the exact figures of -1 and 1 for any natural correlation.
With these considerations, we can conclude that the correlation coefficient between average temperature and coat sales is most likely to be between 0 and -1.
