Math 2, Draw Scatter Plots and Best-Fitting Lines
Math 75, Mod3 Scatter Plots and Best-Fitting Lines Name: ______________________
Worksheet Review (FINISH FOR HOMEWORK)
|SCATTER PLOTS |A scatter plot is a graph of a set of data pairs (x, y). If y tends to increase as x increases, then the data have a positive correlation. |
| |If y tends to decrease as x increases, then the data have a negative correlation. If the points show no obvious pattern, then the data have|
| |approximately no correlation. |
| |[pic] |
|Example 1: |TELEPHONES Describe the correlation shown by each scatter plot. |
| |Cellular Phone Subscribers and Cellular Service Regions, 1995–2003 |
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| |Cellular Phone Subscribers and Corded Phone Sales, 1995–2003 |
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| |[pic] |
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| |A. There is a very strong linear positive correlation between subscribers and cell service |
| |regions. |
| |B. There is a strong linear negative correlation between subscribers and corded phone sales. |
|CORRELATION COEFFICIENTS |A correlation coefficient, denoted by r, is a number from -1 to 1 that measures how well a line fits a set of data pairs (x, y). If r is |
| |near 1, the points lie close to a line with positive slope. If r is near -1, the points lie close to a line with negative slope. If r is |
| |near 0, the points do not lie close to any line. |
| |[pic] |
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|Example 2: |Tell whether the correlation coefficient for the data is closest to -1, -0.5, 0, 0.5, or 1. |
| |[pic] |
|Solution: | |
|a. -0.5 | |
|b. 0 | |
|c. 1 | |
|Example 3: |For each scatter plot, (a) tell whether the data have a positive correlation, a negative correlation, or approximately no correlation, and |
| |(b) tell whether the correlation coefficient is closest to -1, -0.5, 0, 0.5, or 1. |
| |[pic] |
| | |
| |0.5 -1 0 |
|BEST-FITTING LINES |If the correlation coefficient for a set of data is near ±1, the data can be reasonably modeled by a line. The best-fitting line is the |
| |line that lies as close as possible to all the data points. You can approximate a best-fitting line by graphing. |
| |Approximating a Best-Fitting Line |
| |STEP 1 Draw a scatter plot of the data. |
| |STEP 2 Sketch the line that appears to follow most closely the trend given by the data points. There should be about as many points above |
| |the line as below it. |
| |STEP 3 Choose two points on the line, and estimate the coordinates of each point. |
| |STEP 4 Write an equation of the line that passes through the two points from Step 3. This equation is a model for the data. |
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|Example 4: |The table shows the number y (in thousands) of alternative-fueled vehicles in use in the United States x |
| |years after 1997. Approximate the best-fitting line for the data. |
| |x |
| |0 |
| |1 |
| |2 |
| |3 |
| |4 |
| |5 |
| |6 |
| |7 |
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| |y |
| |280 |
| |295 |
| |322 |
| |395 |
| |425 |
| |471 |
| |511 |
| |548 |
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|Solution: Answers will vary. | |[pic] |
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|Extension: |Use the equation of the line of fit from the above example to predict the number of alternative-fueled |
| |vehicles in use in the United States in 2010. |
|Solution: If you are confident the trend will | |
|continue, you could use x=13. Otherwise, 2010 is | |
|outside the scope of the data. | |
|5. |A line that lies as close as possible to a set of data points (x, y) is called the best fit line for the data points. |
|6. |Describe how to tell whether a set of data points shows a positive correlation, a negative correlation, or approximately no correlation. |
| |Positive correlation: As x increases, y increases. |
| |Negative correlation: As x increases, y decreases. |
| |No correlation: No visible pattern appears. |
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|Tell whether the data have a positive correlation, a negative correlation, or approximately no correlation. |
|7. |[pic] |8. |[pic] |9. |[pic] |
| |Negative corr. | |Positive corr. | |Approx. no corr. |
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|Tell whether the correlation coefficient for the data is closest to -1, -0.5, 0, 0.5, or 1. |
|10. |[pic] |11. |[pic] |12. |[pic] |
| |0 | |0.5 | |-1 |
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|In Exercises 13–14, (a) draw a scatter plot of the data using an appropriate scale, (b) approximate the best-fitting line, and (c) estimate y when x = 20. Graph |
|can be found on the last page. |
|13. |[pic] |14. |[pic] |
| |b. answers will vary |b. answers will vary |
| |c. around 3.8 |c. beyond the scope of the data |
| |MULTIPLE CHOICE Which equation best models the data in the scatter plot? | |
|15. |[pic] | |
| |b. (although if your line is off, d. might be match your line) |[pic] |
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|16. |ERROR ANALYSIS The graph shows one student’s approximation of the best-fitting |[pic] |
| |line for the data in the scatter plot. Describe and correct the error in the | |
| |student’s work. | |
| |The line is drawn too low. The sum of the residuals above the line should be | |
| |close to the sum of residuals below the line. | |
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|17. |MULTIPLE CHOICE A set of data has correlation coefficient r. For which value of r would the data points lie closest to a line? A |
| |[pic] |
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|18. |The data pairs (x, y) give U.S. average annual public college tuition y (in dollars) x years after 1997. Find the best-fitting line for the data |
| |using Statcato. Also write the value of r and r squared. |
| |(0, 2271), (1, 2360), (2, 2430), (3, 2506), (4, 2562), (5, 2727), (6, 2928) |
| |Y = 2236.607 + 101.321x |
| |R = .9724 |
| |R squared = .9455 |
| |SE = 57.5549 |
| |The important point is to be able to interpret the meaning of these values in context. |
| |Slope: The average tuition increases $101.32 per year. |
| |Y intercept: The tuition was $2236.61 in 1997. |
| |R: The correlation coefficient r=.9724 shows that there is a very strong positive correlation between year and tuition. |
| |R squared: 94.55% of the variation in tuition can be explained by the year. |
| |SE: One can roughly expect an error of ±$57.55 in tuition when making predictions. |
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