Algebra 2 Notes
Algebra 2 Notes Name: ______________________
Section 2.5 – Curve Fitting with Linear Models, Linear Regression
Researchers are often interested in how two measurements are related. The statistical study of the relationship between variables is called ____________________.
A scatter plot is helpful in understanding the form, direction, and strength of the relationship between two variables. ____________________ is the strength and direction of the linear relationship between two variables.
|Positive correlation |Negative correlation |Relatively No Correlation |
|Positive slope |Negative slope | |
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If there is a strong linear relationship between two variables, a ____________________, or a line that best fits the data, can be used to make predictions. The ____________________ [pic] is a measure of how well the data set is fit by a model.
|Properties of the Correlation Coefficient [pic] |
|[pic] is a value in the range [pic]. |
|If [pic], the data set forms a straight line with a ____________________ slope. |
|If [pic], the data set has no correlation. |
|If [pic], the data set forms a straight line with ____________________ slope. |
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|CAUTION! Don’t confuse slope with the value of [pic]. Whether a line has a slope of [pic] or a slope of [pic], it can have an [pic]-value of 1. The |
|[pic]-value and the slope have the same __________. |
You can use a graphing calculator to perform a linear regression and find the correlation coefficient [pic]. To display the correlation coefficient, you may have to turn on the diagnostic mode.
Example 1: Meteorology Application
|Use your graphing calculator to make a scatter plot for the temperature data, identify the correlation, and find the equation of the line of best fit. |
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|Average High Temperatures ([pic]) |
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|Jan |
|Feb |
|Mar |
|Apr |
|May |
|Jun |
|Jul |
|Aug |
|Sep |
|Oct |
|Nov |
|Dec |
| |
|Akron |
|33 |
|37 |
|48 |
|59 |
|70 |
|78 |
|82 |
|80 |
|73 |
|61 |
|49 |
|38 |
| |
|Wellington |
|67 |
|67 |
|65 |
|61 |
|56 |
|53 |
|51 |
|52 |
|55 |
|57 |
|60 |
|64 |
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Example 2: Anthropology Application
|Anthropologists use known relationships between the height and length of a woman’s humerus bone, (the bone between the elbow and the shoulder) to |
|estimate a woman’s height. Some samples are shown in the table. |
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|Bone Length and Height in Women |
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|Humerus Length (cm) |
|35 |
|27 |
|30 |
|33 |
|25 |
|39 |
|27 |
|31 |
| |
|Height (cm) |
|167 |
|146 |
|154 |
|165 |
|140 |
|180 |
|149 |
|155 |
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|(a) Use your graphing calculator to make a scatter plot for the data with the humerus length as the independent variable. |
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|(b) Find the correlation coefficient [pic] and the line of best fit. Interpret the slope of the line of best fit in the context of the problem. |
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|(c) A humerus 32 cm long was found. Predict the woman’s height. |
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Example 3: Nutrition Application
|Find the following information for this data set on the number of grams of fat and the number of calories in sandwiches served at Dave’s Deli. |
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|Dave’s Deli Sandwiches Nutritional Information |
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|Fat (g) |
|5 |
|9 |
|12 |
|15 |
|12 |
|10 |
|21 |
|14 |
| |
|Calories |
|360 |
|455 |
|460 |
|420 |
|530 |
|375 |
|580 |
|390 |
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|(a) Use your graphing calculator to make a scatter plot for the data with the humerus length as the independent variable. |
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|(b) Find the correlation coefficient [pic] and the line of best fit. Draw the line of best fit on your scatter plot. |
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|(c) Predict the amount of fat in a sandwich with 500 calories. How accurate do you think your prediction is? |
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