1. In a data set with a strong linear correlation, the points in the scatter plot approximate a line.Turn on Show least-squares fit line. The least-squares fit line is the “best-fit” line, or theline that most closely “fits” the shape of the data.A. When r = 1, how are the points in the scatter plot related to the least-squares fit line?B. Slowly decrease r. How does this affect where the points are in relation to the line?
Question
- In a data set with a strong linear correlation, the points in the scatter plot approximate a line.Turn on Show least-squares fit line. The least-squares fit line is the “best-fit” line, or theline that most closely “fits” the shape of the data.A. When r = 1, how are the points in the scatter plot related to the least-squares fit line?B. Slowly decrease r. How does this affect where the points are in relation to the line?
Solution
A. When r = 1, the points in the scatter plot are perfectly aligned with the least-squares fit line. This is because r = 1 indicates a perfect positive linear correlation, meaning all points lie exactly on the line.
B. As you slowly decrease r from 1, the points in the scatter plot start to deviate more from the least-squares fit line. This is because a lower r value indicates a weaker linear correlation. The points will start to spread out more around the line, and the line will no longer perfectly fit the data. The lower the r value, the more scattered the points will be around the line.
Similar Questions
Activity:Correlation andlines of best fitGet the Gizmo ready: Set r to 1.00. (To quickly set a slider to a specificvalue, type the value into the text box to the rightof the slider, and hit Enter.)1. In a data set with a strong linear correlation, the points in the scatter plot approximate a line.Turn on Show least-squares fit line. The least-squares fit line is the “best-fit” line, or theline that most closely “fits” the shape of the data.A. When r = 1, how are the points in the scatter plot related to the least-squares fit line?B. Slowly decrease r. How does this affect where the points are in relation to the line?2. With Show least-squares fit line still selected, set r to 0.90. The points should be close tothe line, but not right on it. Below Generate new data set with: click Same r several times.A. Do all the least-squares fit lines for these scatter plots have the same slope?B. Do all the least-squares fit lines have the same y-intercept?C. What do all the least-squares fit lines have in common?A positive r indicates a positive correlation: as x increases, y also tends to increase.D. Set r to –0.90. Click Same r several times. What do the least-squares fit lines forthese scatter plots have in common?A negative r indicates a negative correlation: as x increases, y tends to decrease.3. Set r to 0.00. Click Same r several times.A. Do all the least-squares fit lines for these scatter plots have the same slope?B. Do all the least-squares fit lines have the same y-intercept?C. What do all the least-squares fit lines have in common?When r = 0, there is no correlation in the data. This means that the value of y doesnot seem to be at all related to the value of x.(Activity continued on next page)Activity (continued from previous page)4. Turn off Show least-squares fit line. Click New r, andsketch the scatter plot to the right.What is the value of r?Turn on Fit a line. Use the slope (m) and y-intercept (b)sliders to estimate the line that fits this data set best.Sketch your line and record its equation below.Equation of estimated line:Check your estimate by turning on Show least-squares fit line. Record the equation for theactual least-squares fit line.Least-squares fit line equation: Was your estimate close?5. Turn off Show least-squares fit line. Click New r several times. For each data set, try to fitthe red line to the data, and then check it by turning on Show least-squares fit line.How does the value of r relate to how easy it is to estimate the least-squares fit line?
3. Set r to 0.00. Click Same r several times.A. Do all the least-squares fit lines for these scatter plots have the same slope?B. Do all the least-squares fit lines have the same y-intercept?C. What do all the least-squares fit lines have in common?When r = 0, there is no correlation in the data. This means that the value of y doesnot seem to be at all related to the value of x
With Show least-squares fit line still selected, set r to 0.90. The points should be close tothe line, but not right on it. Below Generate new data set with: click Same r several times.A. Do all the least-squares fit lines for these scatter plots have the same slope?B. Do all the least-squares fit lines have the same y-intercept?C. What do all the least-squares fit lines have in common?A positive r indicates a positive correlation: as x increases, y also tends to increase.D. Set r to –0.90. Click Same r several times. What do the least-squares fit lines forthese scatter plots have in common?A negative r indicates a negative correlation: as x increases, y tends to decrease.
D. Set r to –0.90. Click Same r several times. What do the least-squares fit lines forthese scatter plots have in common?A negative r indicates a negative correlation: as x increases, y tends to decrease
What do all the least-squares fit lines have in common?When r = 0, there is no correlation in the data. This means that the value of y doesnot seem to be at all related to the value of x
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