Question

use the given data to complete parts (a) and (b).

compute the linear correlation coefficient with the additional data point. the linear correlation coefficient for the five pieces of data is 0.883.
(round to three decimal places as needed.)
comment on the effect the additional data point has on the linear correlation coefficient.
a. the additional data point strengthens the appearance of a linear association between the data points.
b. the additional data point does not affect the linear correlation coefficient.
c. the additional data point weakens the appearance of a linear association between the data points.
explain why correlations should always be reported with scatter diagrams.
a. the scatter diagram is needed to see if the correlation coefficient is being affected by the presence of outliers.
b. the scatter diagram can be used to distinguish between association and causation.
c. the scatter diagram is needed to determine if the correlation is positive or negative.

Explanation:

• For the first part, since the linear correlation coefficient for five data points is given as \(0.883\) (a relatively strong positive correlation), and typically, if an additional data - point follows the general trend (strengthens the linear pattern), it will increase the correlation coefficient. If it goes against the trend (weakens the linear pattern), it will decrease the coefficient.
• For the second part:
• Option A: A scatter diagram helps in visualizing the data points. An outlier can have a significant impact on the correlation coefficient. For example, in a simple two - variable data set, one extreme outlier can pull the correlation coefficient in a particular direction. A scatter diagram allows us to identify such outliers.
• Option B: While a scatter diagram can show an association (a pattern between two variables), it cannot prove causation. Causation requires more in - depth analysis (like experimental design, control of variables etc.), not just a scatter diagram.
• Option C: The sign of the correlation coefficient (\(r\)) already tells us if the correlation is positive or negative (\(r>0\) for positive, \(r < 0\) for negative). A scatter diagram is not the primary tool for determining the sign of the correlation coefficient.

Answer:

• For the first part (assuming the additional data - point follows the trend, but since no new data is given for calculation, we go by the options):
• If the additional data - point is in line with the linear trend (strengthens the linear association), the answer for the first sub - question (based on options) is A. The additional data point strengthens the appearance of a linear association between the data points.
• For the second part: A. The scatter diagram is needed to see if the correlation coefficient is being affected by the presence of outliers.