Final answer:
To determine the best correlation coefficient from the given options, we consider the values with the largest magnitude; here, -0.9 and 0.9 are the strongest. A coefficient of at least 0.71 is necessary for a coefficient of determination of 0.5. Statistical significance is determined by comparing the correlation coefficient to critical values, considering sample size and confidence level.
Step-by-step explanation:
The student has presented four potential correlation coefficients: 0.4, -0.9, -0.4, 0.9. The correlation coefficient reflects the strength and direction of a linear relationship between two variables. A coefficient of 1 or -1 signifies a perfect positive or negative linear relationship, respectively.
To estimate the correlation coefficient that best describes the data, one must consider the context and the hypothesized relationship direction. However, if we are to assess the strength of the relationship alone, we would look for the correlation coefficient with the greatest absolute value. In this instance, the coefficients -0.9 and 0.9 have the highest magnitudes, indicating the strongest relationships, be it negative or positive.
To have a coefficient of determination (which is the square of the correlation coefficient) of at least 0.5, one requires a correlation coefficient of approximately 0.71 or higher. However, the provided data includes correlation coefficients higher and lower than this threshold, which informs us whether the relationship can explain at least 50% of the variability in the data.
Referring to critical values tables, as seen in mathematical texts, can determine the statistical significance of a correlation coefficient based on the sample size and desired confidence level. For example, if the sample size is small (say n<30), one might consult a table of critical values for the Pearson correlation coefficient to conclude the statistical significance of the observed coefficient.