Final answer:
Correlation indicates the relationship between variables, measured by the correlation coefficient, and a statistical significant correlation is determined through hypothesis testing. A positive correlation means both variables increase together, while a negative correlation means one decreases as the other increases. The strength is indicated by the magnitude of the correlation coefficient.
Step-by-step explanation:
A correlation refers to the relationship between two or more variables. When variables are correlated, changes in one variable are associated with changes in another. The strength and direction of this relationship are measured by a statistic called the correlation coefficient, typically represented by the letter r, which ranges from -1 to +1. A value of +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and a value of 0 means there is no correlation.
Statistical significance in correlation is determined through hypothesis testing, where we compare the calculated correlation coefficient against a null hypothesis, which usually states that there is no correlation between the two variables (r = 0). If the p-value from this test is less than the chosen level of significance (e.g., 0.05), the correlation is considered statistically significant, implying that the relationship observed is unlikely to have occurred by chance.
A positive correlation means that as one variable increases, the other variable also increases, whereas a negative correlation means that as one variable increases, the other decreases. The magnitude of the correlation coefficient indicates how strong the relationship is. For example, a coefficient close to +1 or -1 means a strong relationship, whereas a coefficient close to 0 indicates a weak relationship.
When determining if the correlation is significant, we might use a t-test to test the hypotheses. For instance, with a correlation of 0.33 in a sample of 30 cases, and a significance level set at 0.05, we would calculate the t-value and compare it with a critical value from the t-distribution to determine significance.